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154 Douglas M. Jesseph

Thomas

Hobbes is perhaps the most

famous

circle

squarer in

history.

Even

those

who know relatively little about his

mathematical

work are

well aware that he published numerous

quadratures.

It is also

widely known

that

Hobbes's

efforts were

refuted by the

British mathematician

John Wallis, and the most salient result of the

long-running

dispute between Hobbes and Wallis was the demolition

of Hobbes's

mathematical reputation.

Less

well-known, however, is

the fact that

Hobbes's

first mathematical publication was part of a

refutation of the

circle-squaring efforts

of the

Danish astronomer-

mathematician

Christian Severin, better known as

Longomontanus.

In a series of publications over

several

decades,

Longomontanus

claimed to have achieved the

long-sought

quadrature

of

the circle (1).

The

mathematician

John

Pell

attacked

his efforts

[Pell,

1644],

but

Longomontanus refused to give

up

the case.

That peevish

Dane

Severin, as

Sir Charles

Cavendish called him (2), promptly objected

to Pell's methods

in

a further pamphlet:

Rotundi in

piano seu

circuli

absoluta

mensura [Longomontanus,

1644].

Pell

prepared a response

by soliciting proofs of the

key

lemma he needed to refute

Longomontanus

from European mathematicians,

evidently

hoping to show that

the basic

result he needed

could

be had

by

different methods and

thereby

place his refutation beyond serious doubt.

Through

the agency

of

Cavendish,

Hobbes

was

asked

for

and

contributed

a

proof

of

the

relevant

result.

This placed him in the

company

of such luminaries

as Roberval, Cavalieri, Mersenne, and Descartes when Pell

published

Controversiae

de vera circuli

mensura... Prima Pars [Pell,

1647].

This

work contains a review of the controversy,

with letters and

demonstrations supplied by ten

"notable mathematicians" in

support of Pell.

The irony here is quite remarkable: the philosopher who

would

publish

and obstinately

defend

more

than

a

dozen

circle quadratures

over

a

period

of

more

than

twenty

years published

his

first

mathematical

piece as part of a campaign to silence an old circle squarer (3).

(1)

These publications

include

[Longomontanus, 1612; 1634; and 1643].

(2) The

letters from

Cavendish

to Pell

are preserved

in

Pell's

papers (British

Library,

Add.

Ms.

4278 and 4280). This reference to Longomontanus

appears

in

a letter of

20 December, 1644 (Add.

Ms.

4278, f.

188r). See [Hervey, 1952] for

an account

of

the

Pell-Cavendish correspondence as it relates to Hobbes and Descartes.

(3) Hobbes's demonstration

survives

in manuscript as British Library, Add.

Ms.

4278,

f. 200r; Cavalieri's proof is Add.

Ms.

4278, f. 251r and is

reprinted

in [Cavalieri, 1987,

no. 94]. See [Jacoli, 1869]

for a

bibliographic

overview

of

the dispute, especially as it

relates

to

Cavalieri.

The theorem

Hobbes

proves

can be

stated

as follows:

if A

is the

tangent to

an

arc less

than

a

quadrant

of

the

circle,

and

В

is the

tangent to

one half

of

the same

arc,

and the circle has radius r, then

A:B::2r

:(r -B

).



Hobbes on the

Methods

of Modern Mathematics 155

I

mention

this

detail

for

two

reasons:

first

to

make

the

point

that

Hobbes's

mathematical

work

does not consist exclusively

of

misguided

efforts to

square the circle

and second to

pose the

question

that looms large

in

any study of Hobbes and seventeenth-

century mathematics: How did a man whose mathematical

ability

was once ranked with the

best

in Europe end up publishing

quadratures

that remain

paradigms of geometric

ineptitude?

I

cannot

give a complete

answer

to this question, but I

can

tell

part of

the

story. My

strategy

is to begin with

a sketch of some of the

important

developments and controversial issues in seventeenth-

century

mathematics,

and

then

turn to

a

brief

exposition

of

Hobbes's

philosophy

of mathematics.

With

this background

in hand

I

will

explore some

of

the

themes

in the Hobbesian mathematical

corpus, concentrating on Hobbes's attitude toward

analytic

geometry

and

the

method

of

indivisibles.

I contend that by

understanding

Hobbes's

relationship to these two

methods

we

can

gain

important

insight

into

his mathematical

work

and

the

context in

which

it

developed. An

understanding

of these

issues

should

then

help

to explain why

Hobbes's

mathematical career took

such an

unfortunate

turn

into

the

hopeless

pursuit

of

unobtainable

results.

As we will see, his program for

mathematics

was inspired by an

ideal of demonstration which led him to underestimate the

difficulty

of important

mathematical problems

while

simultaneously

misunderstanding and

dismissing new

developments.

But Hobbes's

conception of

mathematics and his responses to

the

mathematics

of the seventeenth century are not wholly incomprehensible, nor

are they altogether devoid of

interest.

Hobbes did indeed meet

with disaster in his

campaign

for mathematical glory, but it is

a

disaster

worth studying

for

the

light

it

can

shed

on

his

metaphysics

and

his

sources.

I.

—

The

Mathematical Background

Hobbes's

life spanned an enormous transformation in

European

mathematics. When he was

born

in 1588

the study of

mathematics

centered

on

the

classical

works

of

Greek

antiquity

and

most

of the mathematical work of the

period

was devoted to producing



156 Douglas M.

Jesseph

editions and commentaries

on classical authors. By the

time

of

Hobbes's

death

in

1679 the mathematical landscape had changed

dramatically:

Newton

had long

since

circulated his first papers

on

the

method

of fluxions, Leibniz

had

applied himself

to

mathematics and the great advancement now

known

as the calculus was

well

underway. Two

of the most significant

mathematical

changes

during this

period,

and

the two

most important for an

understanding

of

Hobbes's

mathematical work, were the development of analytic

geometry and the

rise

of infinitesimal techniques, particularly the

"method of

indivisibles."

Indeed,

it is

no exaggeration

to

say

that

the

advent

of these two

methods

marks

the beginning of

modern

mathematics. My aim in this section is to outline the fundamental

ideas

behind

these

mathematical

methods

as

well

as

the

ontroversies they provoked.

1 / The Analytic Art and the

Status

of Geometry

Analytic

geometry

is commonly

taken

to begin in

1637

with

the publication of Descartes's Géométrie, and most

regard

this as

a significant

mathematical

advance.

Hobbes,

however, disparaged

the

analytic

approach

and saw

it

as

something

akin

to

a

mathematical perversion: in

his estimation, the "modern analytics"

had

corrupted geometry and were a significant step backward. The

fundamentals

of

the analytic approach

are familiar

enough that they

need

no detailed account here

(4).

The

essential

point

is that

algebraic operations (addition,

subtraction, multiplication,

division, and

the extraction of roots) are interpreted as geometric constructions.

Then, techniques for

solving

equations

are applied

to

the

investigation of geometric curves. François Viète's Isagoge in Artem

Analyticem [Viète, 1646] is a key

text

in the development of analytic

geometry, most notable

for its expansion

of

algebraic

techniques and their

application

to geometric problems (5). In

Britain,

(4) See

[Boyer, 1956] for

the

standard

account

of

the development

of

the subject,

especially

Chapter 5. Historians

of mathematics

also credit Fermat with the invention

of

analytic

geometry,

but his

work

is not prominent in Hobbes's

polemics

against analytic methods.

I will therefore

ignore

Fermat's

contributions

in the course

of

this study.

(5)

Indeed, Viète's

work

is so fundamental

to

the development

of modern

algebra and

geometry

that Jacob

Klein

called him the true founder

of modern

mathematics [Klein,

1968, 5].



Hobbes

on the Methods

of

Modern Mathematics

157

William

Oughtred's

Clavis Mathematicae

[Oughtred, 1631]

introduced a

generation

of mathematicians to algebraic

methods

and

their use in geometry. Although it is difficult

to

discern the extent

to

which

Descartes is indebted to

Viète,

Oughtred, or

any one

else,

there

is no

question that

all

three

authors promoted

the

incorporation of algebraic

techniques

into geometry (6). This fusion of

algebraic and

geometric

methods allows systematic and

relatively

simple solutions to problems which require

elaborate

constructions

with

compass

and rule in

classical

geometry. Moreover, analytic

geometry can classify curves by

their

characteristic equations and

study

curves

which are more complex than

those

accessible to

classical

investigation.

The role

of

algebraic analysis in the

new

geometry

is summed

up in

Descartes's remark

in

Book II of the

Géométrie:

"Je

pourrois

mettre icy plusieurs

autres moyens

pour

tracer

&

concevoir

des lignes courbes,

qui seroient

de plus

en

plus

composées

par

degrés

a

l'infini. Mais pour comprendre

ensemble

toutes celles, qui sont en la

nature, & les distinguer

par

ordre en

certains

genres, je ne sçache rien

de meilleur

que

de dire

que

tous les poins, de celles qu'on peut nommer

Géométriques, c'est

a

dire

qui

tombent

sous

quelque

mesure

précise

&

exacte, ont nécessairement

quelque rapport a

tous les

poins

d'une

ligne

droite, qui

peut estre exprimée par quelque

equation,

en

tous

par une

mesme..." [AT, 6, 392.]

The Cartesian

program

for

geometry

classifies as

properly

geometric (as

opposed

to mechanical )

any curves which

have a

"precise and

exact"

measure. This rather vague criterion is

elucidated

slightly by Descartes's declaration that geometric curves are

those

which can be described by a regular

motion

or series of

motions

[AT,

6,

390].

Descartes implicitly assumes that any curve

(6) Descartes is famously

reticent

about

sources

for

his mathematical

work

and never

acknowledged

a significant debt to other

mathematicians, as

improbable

as this

may seem.

For

purposes

of our

investigation, the best

statement of Descartes's attitude

comes from

a letter of

Pell

to

Cavendish

dated 12

March,

1646

and reporting

a

conversation with

Descartes

on

mathematical

matters.

Pell writes: "I

perceive

he

demonstrates

not willingly.

He sayes

he

hath

penned

very few demonstrations

in his

life

(understand

after ye style

of ye old

Grecians which he

affects not)

THAT he

never had an Euclide of

his

owne

but

in

4

day

es, 30 yeares agoe... Of

all

ye Ancients

he

magnifies none but

Archimedes,

who

he

sayes, in

his

bookes de Sphaera & Cylindro and a piece or two

more, shows

himselfe

fuisse

bonum Algebraicum &

habuisse

vere-magnum ingenium.

I

will

not

trouble you

of

what

he

said

of

Vieta,

Fermat

and Roberval and

Golius:

Of

Mr.

Hobbes

I

durst

make

no mention to

him.

(British Library, Add.

Ms.

4280, f.

117r.)

The letter is partially

reprinted in [AT, 4, 729-732] and fully reprinted in

[Hervey,

1952,

77-79].




representable

algebraically

could be traced

by such

continuous

motions and

therefore admitted

as

geometric,

and

conversely

that

any

geometric curve

could

be

represented by an equation

(7). In

the solution of

problems,

the choice of curves

is constrained

by

the requirement that the simplest curve (/. e., the curve whose

equation is

of

lowest

degree)

be employed. Hobbes

rejects

this

reliance

upon

algebraic or

equational criteria

for

the

classification

of curves

and

the solution of

problems,

for

he

holds

that

such

techniques

have introduced

a bewildering array of empty symbols into

geometry

and

distracted geometers from

their true business —

the

construction

and study

of geometric

magnitudes.

Another

important difference between

analytic and

classical

methods

concerns

the

manner

in

which

algebraic

operations are

interpreted

in geometry. Classically,

the

geometric multiplication

of two lines yields a

rectangle,

or the product of

three

lines a

solid.

But

Descartes interprets multiplication as

an

operation

which

leaves the dimension of the

product

homogeneous

with that

of

the

multiplicands.

Just as the product of two numbers is

a

number,

Cartesian

analytic geometry treats the product

of two

lines as a

line.

And

in general, all operations in analytic

geometry are

operations

on line segments which

result in

new line segments. This

is

the

import

of

Descartes's

declaration

at

the

beginning of

the

first

book of

the Géométrie:

Tous

les Problesmes de Geometrie se peuvent facilement réduire a

tels termes, qu'il n'est besoin par

après

ques

de connoistre

la

longueur

de

quelques

lignes droites, pour les construire. Et

comme

toute l'Aritme-

tique

n'est

composée, que de quatre

ou

cinque operations,

qui

sont

l Addit ion, la

Soustraction,

la

Multiplication,

la Division, & Extraction des

racines,

qu'on

peut prendre pour une espèce de Division : Ainsi n'at'on

autre chose

a

faire en Geometrie touchant les lignes qu'on cherche, pour

les préparer

a

estre connues, que leur en adjuster d'autres, ou en

oster,

Oubien

en

ayant

une,

que

je

nommeray l unité pour

la

rapporter

d'autant

mieux

aux

nombres, &

qui

peut ordinairement estre prise a discretion,

puis en

ayant

encore deux autres, en

trouver

une quatriesme,

qui

soit

à

l'une

de ces deux, comme l'autre

est a l'unité,

ce qui

est le

mesme

que la Multiplication;

oubien en trouver

une quatriesme,

qui

soit à

l'une

(7) See [Bos,

1981] for a

discussion

of

Descartes's program

for

geometry and the

difficulties surrounding his classification and representation

of

curves. As Bos notes, Descartes

could

not simply take as geometric all curves that admit an algebraic equation;

if

he were to adopt this criterion, Descartes could no longer claim that he was doing

geometry

[Bos, 1981,

305].

The result is that the use of a purely

algebraic

criterion for

geometric curves is

merely

implicit in Descartes's work.



Hobbes

on the Methods

of

Modern Mathematics

159

de

ces

deux,

comme

l unité

est a

l'autre,

ce

qui

est

le

mesme

que

la

Division;

ou

enfin trouver une, ou deux, ou

plusieurs moyennes

proportionnelles

entre

l'unité, & quelque autre ligne; ce

qui

est le mesme

que

tirer la racine quarrée,

ou

cubique, etc. Et ie ne craindray

pas

d introduire

ces termes

d'Arithmétique

en la

Geometrie,

affin de

me

rendre

plus intelligibile. [AT,

6, 370.]

This

conception of geometry

is

underwritten by a strong

thesis

on

the unity

of arithmetical

and geometric magnitudes. Descartes

sees

nothing peculiarly arithmetical about

the

operation of

addition, or

anything

uniquely

geometrical

about the extraction of roots.

The

resulting

application

of

algebra

to

geometry

therefore

treats

algebra

as

a science of

magnitude in

general, and

the

specifically

geometric content of a

problem is removed

(and, at least on Des-

cartes's view, the problem is rendered

more

intelligible) when

it

is

represented

as a relation among various abstract magnitudes.

A recurrent theme

in Hobbes's mathematical

work is

the inappro-

priateness of this

conflation

of geometry and arithmetic through

the mediation

of algebra.

Hobbes argues that, far from making

geometry

more

intelligible, the

use

of algebra

is

a source of

confusion

and

error.

The advent of analytic

methods

provoked

a philosophical

debate

on the

question

whether arithmetic or geometry was the genuinely

foundational

discipline in mathematics

(8). Classical

mathematicians

distinguished

discrete quantity

( number ) from

continuous

quantity ("magnitude"),

declaring

the

former to

be the object of

arithmetic and

the

latter to be

the

proper

object of

geometry.

Classically, then,

geometry

and

arithmetic

are

distinct

sciences with

no

common

object, so

there

is no need to ask which is the more

fundamental science. This

situation

changed with

the development

of analytic

geometry.

Many

interpreted

algebra

as

a

kind of

generalization of

arithmetic, and it was often

characterized

as the

arithmetic of

species," in

which variables such as x or a were

taken

as general

representatives

of kinds or species of

quantities.

In this

scheme,

the

basic principles of algebra were

seen as

deriving

from

arithmetic,

and

the

prominence

of

algebraic methods in analytic

geometry

led

some

to conclude that geometry must, in

some

important sense, be based on arithmetic.

(8) See [Pycior,

1987]

and

[Sasaki, 1985] for

other

studies of

this

debate.



160

Douglas

M.

Jesseph

John

Wallis

argued for the primacy

of

arithmetic over

geometry

in

his 1657

Mathesis

Universalis. Wallis was appointed

Savi-

lian

Professor

of Geometry at Oxford

in

1649

and

the Mathesis

Universalis apparently

began as

Savilian

lectures. The work

contains a general statement

of

his philosophy

of

mathematics

which

marshals

philosophical, historical,

and

philological arguments

for

the

claim

that arithmetic is the true foundation

of

all

mathematics. Indeed, Wallis's point of view

is

evident

in

the full title of

the

Mathesis Universalis,

which

promises

(among

other things)

a

complete Arithmetical work, presented both philologically and

mathematically, encompassing

both the

numerical as well as

the

specious

or symbolic

arithmetic,

or geometric calculus (9).

As

part

of

this

program

to

elevate

arithmetic

to the

status of

universal mathematics," Wallis devotes the

twenty-third

chapter

to

a series

of "arithmetical"

demonstrations

of results from the

second

book of Euclid's

Elements, with

the

intent

of

showing that

the

important results in geometry

can

be

obtained

more quickly and

easily

by

employing arithmetical principles. In his

philosophical

case

for

the primacy of arithmetic, Wallis

admits that

such

geometric

terms as

root, square,

and cube

appear

in

algebra,

but

denies that this

should lead

to

the conclusion

that

algebra

is

based

on

geometry.

Although some authors have

drawn

this

conclusion, Wallis claims that

geometry

ultimately

takes

its principles

from arithmetic

(10).

Part

of

his reasoning is

the argument

that

universal

algebra is

fundamentally arithmetical

and not

geometrical.

He insists:

"Indeed many geometric things can be

discovered

or elucidated by

algebraic

principles,

and

yet it does not

follow

that

algebra is

geometrical, or even that it is based on geometric principles (as some would

(9)

The

full

title

reads,

Mathesis

Universalis:

Sive,

Arithmeticum

Opus

Integrum,

Turn

Philologice, turn Mathematice tràditum, Arithmeticam turn

Numerosam,

turn Speciosam

sive

Symbolicam complectens,

sive

Calculum Geometricum

;

turn etiam Rationum

Propor-

tionumve

traditionem ; Logarithmorum item

Doctrinam; aliaque, quae

Capitum

Syllabus

indicabit.

(10)

Because some take the

geometric

elements for the

basis

of

all

of mathematics,

they

even think

that all

of arithmetic is to be

reduced

to geometry,

and that there

is no

better

way to show the truth of arithmetical theorems

than

by

proving them from

geometry. But in fact arithmetical

truths

are of a

higher

and more

abstract

nature than those

of geometry. For

example,

it is not because a two foot

line

added to a two foot

line

makes a

four

foot

line

that two and two are four, but

rather

because the latter is true,

the

former

follows [MU, 11, 53].



Hobbes on the

Methods

of Modern Mathematics

161

seem

to

think).

This

close

affinity

of

arithmetic

and

geometry

comes

about,

rather, because geometry

is as it were subordinate to arithmetic,

and

applies universal principles of

arithmetic to its special objects. For,

if

someone

asserts that a line of three

feet added

to a line of two

feet

makes a line five feet long, he asserts

this because the

numbers

two and three added

together

make

five;

yet

this calculation is not

therefore

geometrical,

but clearly arithmetical,

although

it

is used in

geometric measurement. For the assertion of the

equality

of the

number

five

with the

numbers

two and

three

taken

together is a

general

assertion, applicable to other kinds

of

things

whatever,

no less

than

to

geometrical

objects.

For

also

two angels and three angels make five

angels.

And

the very

same reasoning holds of

all arithmetical and

especially

algebraic

operations,

which

proceed

from

principles more

general than those in geometry, which are restricted to

measure."

[MU,

11, 56.]

Isaac

Barrow,

the first

Lucasian Professor

of Mathematics at

Cambridge,

rejected

this

reasoning.

In fact, he tried to turn the

tables

on Wallis by

arguing that

geometry

is

ultimately the

foundation of all

mathematics.

In the third of his Mathematical

Lectures

Barrow considers Wallis's argument for

the

priority

of

arithmetic

and

issues

the

following rebuttal:

To

this I respond by asking How

does

it

happen

that

a

line of

two

feet added

to a

line

of two palms

does not

make a line of four

feet, four palms, or four of any denomination, if it is

abstractly, i. e.,

universally and absolutely true

that two

plus

two

makes four. You

will

say, this is

because

the

numbers are

not applied to the same matter or

measure. And I

would say the

same thing,

from

which I

conclude that

it is not from the

abstract

ratio of

numbers

that two and two make

four, but from the

condition

of the matter to which they are

applied.

This is

because any

magnitude denominated by

the

name

two added

to

a

magnitude

denominated

two

of

the

same

kind will

make

a

magnitude

whose denomination will

be four. Nor indeed

can anything more

absurd

be imagined than to affirm that the proportions of magnitudes to one

another

depend

upon the

relations

of the

numbers by

which they may

be expressed." [LM,

3,

53.]

Barrow's case for the primacy of geometry hinges on the claim

that numbers,

in

and of themselves, are mere symbols whose

content derives from

their

application

to

continuous geometric

magnitude. To put

it

another way,

there

are

no

"numbers

in

the

abstract"

to

serve

as

the

object

of arithmetic, except those

which arise from the

consideration

of homogeneous magnitude



162

Douglas M.

Jesseph

and

its

division

(11).

Barrow goes

to

the extreme

of denying

that

algebra

is

a

mathematical

science at all, classifying

it as

simply part of logic or a set of rules for manipulating symbols.

He distinguishes

two

branches

of algebra,

analytics and

logistics.

The

former, he says

"seems

to be

no

more proper to

mathematics than

to

physics, ethics,

or

any other

science.

For

it

is only

a

part

or species

of

logic." The latter is

no part of

mathematics

because it

has no object

distinct and proper to itself, but only

presents a kind of

artifice, founded

on geometry

(or

arithmetic),

in

which

magnitudes

and numbers are designated by

certain notes

or symbols, and

in which

their sums

and differences are collected

and

compared." [LM, 2, 46.] The

difference

of opinion between

Barrow and

Wallis

is

significant

for

understanding

Hobbes's

relationship

to seventeenth-century

mathematics.

Hobbes

developed

views

on

this particular

question

which

are close

to

Barrow's,

and his

account

of the

nature

of

mathematics

becomes

more

intelligible if

we

see

him as responding to the concerns

which

produced this

dispute over

the relative

priority

of arithmetic

and

geometry.

2

/

The

Method

of

Indivisibles

The development of the

method

of indivisibles was another

pivotal

episode

in seventeenth-century mathematics,

notable both

for providing

a wealth of new

results

and its

share

of controversy.

The first exposition of the

new

method

was

in

Bonaventura

Cavalieri's

Geometria

indivisibilibus continuorum

nova

quadam

ratione promota

[Cavalieri,

1635]. The

method

plays

upon

the

intuition that we

can

reason about the area of a figure by considering

the

lines

it

contains, which

Cavalieri

calls

the

indivisibles of

the

(11)

"I say that mathematical number is not something

having

existence

proper

to itself,

and

really

distinct

from

the magnitude it

denominates,

but is

only

a kind of note or

sign

of magnitude considered in a certain manner; so far

as

the magnitude is

considered as

simply

incomposite,

or

as

composed out of certain

homogeneous

equal parts, every one

of

which

is taken simply and denominated a

unit...

For in order to expound and

declare

our conception of a magnitude, we designate it by the

name

or character of a certain

number, which

consequently

is nothing other

than

the

note

or symbol

of such magnitude

so taken. This is the general nature, meaning, and account of a mathematical number

[LM,

3, 56].

See [Mahoney,

1990,

186-189] for more

on Barrow's account

of number.



Hobbes on the

Methods


figure (12).

Cavalieri

was

cautious

about

claiming

that

these

indivisibles actually compose the figure,

although

he did seek

analogies between the

composition of cloth

out

of threads

and the

relationship

of

a figure to

its

indivisibles

(13).

Instead

of simply

taking indivisibles

as infinitely

small components of finite

magnitudes,

Cavalieri sought

to

introduce

indivisibles as a

new

species

of

magnitude

which could be brought

within

the purview of the

classical

theory of ratios.

К

L

~~

IF

с

-M

Fig. 1. — From B. Cavalieri,

Exercitationes Geometricae Sex, op.

cit., p. 4.

In Cavalieri's

terminology,

all

the

lines"

of the plane

figures

ABCD

and

EFGH

in

figure 1 are produced by the transit of

the

line

LM (called the régula) through the figures. Significantly,

Cavalieri avoids the question

whether there

are an

infinite

number of

indivisibles

produced

by the transit of the

régula

LM or whether

these indivisibles

are infinitely

small when compared with

the

figures,

apparently hoping that his method

would be acceptable on any

resolution

of the problems surrounding the

infinite.

He speaks

vaguely of

an

"indefinite"

number of

lines

contained

within

a

figure, and stresses that the ratios can be compared either

collectively (as one

collection of indivisibles

to another), or

"distribu-

(12)

In a

perfectly

analogous manner the motion of a plane through a

solid

could

produce

all

the

planes

of the solid, or the indivisibles of the solid. For more on Cavalieri

and

his

method see [Andersen, 1985], [De Gandt, 1991], and [Giusti, 1980].

(13) In

his Exercitationes Geometricae Sex,

he declares:

"It is

manifest

that we

can

conceive of plane figures in the

form

of cloths

woven

out of parallel threads, and

solids

in the form

of

books,

which are

built up

out of

parallel pages. He nevertheless

quickly

adds:

But

the

threads

in

a

cloth

and

the

pages

in

a

book

are always

finite

and

have

some thickness, while in this

method

an

indefinite

number of lines in plane figures

(or

planes

in solids) are to be supposed, without any thickness

[Cavalieri, 1647, 3-4].



164

Douglas M.

Jesseph

tively"

by

comparing corresponding

lines

singly

(14). His

appeal

to continuous motion arises from

similar

concerns: he

seems

to

have regarded

this

as

a

relatively

unproblematic concept

which

can

sidestep thorny questions

concerning

infinity. After

all,

anyone will

admit that a

line can

pass through the figure, and the

intersection

of the line

and figure

will

produce, in some

fairly

innocuous sense,

all the lines" contained

in

the figure.

Given this starting

point Cavalieri treated

all the lines"

of

the figure as a

new

species of geometric magnitudes which

could

be dealt with according to

the theory of

magnitudes

in

Book V

of Euclid's Elements. His strategy

is

to

establish

a ratio between

the indivisibles of two figures (either distributively or collectively),

and

then

to

conclude

that

the

same

ratio

holds

between

the

areas

of the figures. This is the

import

of the

"very

general

rule" he

announces

in

the first of his

Exercitationes

Geometricae Sex:

"From these two

[ways of comparing

indivisibles]

a

single and most

general rule

can

be fashioned, which

will

be a summary of all

this

new

geometry, namely this: Figures, both

plane

and solid,

are in

the same

ratio as that of

their

indivisibles compared with one another

collectively

or...

distributively." [Cavalieri, 1647, 6-7.]

Cavalieri's evident caution on foundational matters was not

shared

by other mathematicians of the

seventeenth

century,

most

notably

Wallis (15).

In Wallis's treatment, geometric problems

are

represented

analytically

and

solved

by

determining

the relationship

between the infinite sums

of

infinitely small indivisibles

which

compose

the figures.

As

an

example, consider his approach to

the

quadrature of

the

cubic parabola

in

his

Arithmetica

Infinitorum.

He begins

with

arithmetical results in Proposition 39, observing that:

(14) These two different presentations

of

the

method

appear

more

clearly in his 1647

Exercitationes Geometricae Sex, although the second is also

contained

in the last book

of

the Geometria. He explains the distinction between the two

procedures

thus: The first

method

proceeds by the first kind of reasoning, and compares aggregates of

all

the

lines

of a figure or all the

planes

of a

solid

to one

another,

however

many

they may be. But

the second

method

uses the second kind of reasoning, and

compares single lines

to

single

lines and single planes to single planes, lying in the same direction

[Cavalieri, 1647,

4].

(15) [Andersen, 1985,

Section

10], [Giusti,

1980,

40-65], [Jesseph, 1989] and [Wallner,

1903]

discuss various

reactions

to Cavalieri's

method

and the various changes in the

fundamenta l

concepts. For our purposes, the

most significant

feature

of

Wallis's reaction is

his use

of

infinite

sums of

infinitely

small

parallelograms where Cavalieri had relied upon

finite ratios

of

"all

the

lines

of

one

figure

compared with

another.



Hobbes

on the

Methods


165

0+1

1

2

1 1

______

___

__ __ __

i_

1 + 1

~

2

~

4

~

4 4

0+1 + 8 9 _ 3 _ 1 1

8 + 8 + 8~24~8~4 8

0+1+8 + 27 36_4_ l

27 + 27 + 27 + 27 108 12 4 12

0+1 + 8 + 27 + 64

100

_ 5 _ 1 1

64 + 64 + 64 + 64 + 64 320 16 4 16

From

these initial cases, Wallis concludes "by induction" that

as

the number of

terms in

the

sums increases, the

ratio approaches

arbitrarily near to the

ratio

1:4.

Proposition

41,

which he

takes

to

follow obviously

from

Proposition 39 asserts

that:

If an infinite series is taken

of quantities

in triplicate ratio to

a

continually

increasing

arithmetical progression, beginning with 0 (or, equi-

valently,

if a

series

of

cube

numbers is taken) this

will

be to the

series

of numbers equal to the

greatest

and equal

in

number as one to four."

[AI,

41,

382-383.]

Given this result, Wallis

turns

to the quadrature of the cubic

parabola in

Proposition

42, treating it as an infinite sum of lines forming

a series of cubic

quantities as in

figure

2:

"And indeed let AOT

(with

diameter

AT,

and corresponding

ordi-

nates TO, TO, &c.) be the complement of the cubic

semiparabola

AOD

(with

diameter

AD and

corresponding ordinates

DO, DO, &c).

Therefore,

(by Proposition

45 of

the

Treatise

of Conic Sections) the

right lines

DO, DO, &c. or

their

equals AT, AT, &c. are in subtriplicate ratio of

the

right

lines

AD,

AD

&c.

or

their equals

TO, TO,

&c.

And

conversely

these

TO, TO,

&c. are

in

triplicate ratio of the right lines

AT, AT,

&c. Therefore the whole figure

AOT

(consisting of the infinity of right

lines

TO, TO,

&c.

in triplicate

ratio of the arithmetically proportional

right

lines AT, AT, &c.)

will be to

the parallelogram

TD (consisting

of just as many lines all equal to

TO)

as one to

four.

Which was to

be

shown.

And

consequently

the

semiparabola

AOD (the residuum of

the parallelogram)

is to

the parallelogram

itself

as one

to four." [AI,

42, 383.]

Here, Wallis

takes

the figure as literally composed of indivisibles

and

unhesitatingly

applies

arithmetical principles

to the

solution

of geometric problems. The

"induction"

that leads to his

main



166

Douglas M. Jesseph

Fig.

2.

— From

J . Wallis,

Arithmetica

Infinitorum, op. cit., p.

383.

arithmetical

result

is clearly not consonant with the

classical

standard of rigor,

nor

is his procedure of taking a

ratio

between

two infinite series.

But

note

also

that his

use

of the

method

of

indivisibles departs

from the

classical

approach to geometry because

it fails to observe

the

distinction between

discrete

and

continuous

magnitudes. In treating

a

continuous

geometric figure

as

composed

of

sums

of

discrete

points or lines, the method of indivisibles

simply

ignores

the classical distinction.

Such

departures

from the

classical

standard prompted serious

debate among seventeenth-century mathematicians over the

rigor

and

reliability of the method.

In particular,

Paul Guldin

attacked

the

method as ill-founded and unreliable, and his

Centrobaryca

[Guldin, 1635-1641]

contained

a long polemic

against

Cavalieri in

which he argued that the

method

of indivisibles offends against

the

principle

that

there

can

be

no ratio

between infinities. As he

observes,

the attempt to find a

ratio

between all the lines" of

two figures

can only

be understood as an attempt to compare

one

infinite totality with

another

— but this is explicitly barred by



Hobbes on the Methods of Modern

Mathematics

167

the

most

basic

canons

of

mathematical

intelligibility

(16).

Guldin

was not alone in his reservations

about

the

method:

others, including

Galileo and the

Jesuit

mathematician André Tacquet found fault

with it and argued that it

could lead

to

false

results

(17). Even

the

proponents

of the new method admitted

that it

must be applied

with

caution

in

order to avoid paradox. Indeed, one of

the key

projects in

seventeenth-century

mathematics

was to sort out the

conditions

under which

the method

of

indivisibles could

be used,

and ultimately to render

it

as

secure

as

classical

methods. Hobbes

was

familiar both

with the

method

and

the

controversies

surrounding it, and

one of the most important tasks of this

essay

will

be

to

clarify the relationship between

Hobbes' s

own approach

to

mathematics and

the method of

indivisibles.

II. — Hobbes's Philosophy of

Mathematics

Hobbes's

philosophy

of mathematics

is

grounded

in

his

strict

materialism. He is well known for

denying

the

existence of

immaterial spirits and insisting that all

the phenomena of the

world

must

be

accounted

for in terms

of bodies

and their motions.

Extended

to

the

philosophy

of

mathematics, this metaphysical

program

rules

out

a Platonism which locates

mathematical

objects

in

a supramundane realm of pure forms.

But

it

also

bars

a

typically Aristotelian

conceptualism

that declares mathematics

to

be

a science of

abstractions,

whose objects are formed by

the

intellect

and

reside

in

the

rarefied

(immaterial)

world of

the

pure

understanding.

In the Hobbesian scheme,

only bodies

exist and it is

only through their

motions

that anything can come about.

Thus,

Hobbes

faces

the formidable task of

explaining

how the language

(16) Guldin objects: All

the

lines and all

the

planes

of one

and

another figure are

infinite and

infinite; but

there is no

proportion or

ratio

of an

infinite

to an infinite.

Therefore,

etc. Both

the major and

minor

premises

are clear to

all

geometers,

and so do not

need

many

words [Guldin,

1635-1641,

4, 341]. For

more on Guldin's critique,

which

included the charge that Cavalieri had plagiarized his method from Kepler, see

[Andersen,

1985, Section 10] and [Giusti,

1980,

Section

3].

(17)

Galileo's

objections

appear

largely in

letters

to

Cavalieri,

which

are

studied

in

[Andersen,

1985],

[Giusti,

1980],

and [De Gandt, 1991]. Tacquet's objections

are presented

in

the essay Cylindrica et

Annularia.

See [Tacquet, 1668,

vol. 3,

38-39].



168

Douglas

M.

Jesseph

of

mathematics

can

be interpreted exclusively

in

terms of the

fundamental

principles of

body. In other words, Hobbes

must

seek

a

reduction of

all

mathematics

to the

concepts of motion

and

extension —

the two

fundamental

principles of

body.

In

working

out his philosophy of

mathematics

Hobbes

takes

geometry

to

be

more

basic than arithmetic

or

algebra, since the

concepts of motion and extension

have

a relatively straightforward

geometrical

content.

He

thus

shares Barrow's opinion

on

the

classification of

the mathematical sciences:

geometry

is

the

founda-

tional

science and is

concerned with

the

determination

of quantity,

while arithmetic must ultimately be based on geometric

considerations.

The

quantities considered

by

geometry

are continuous

magnitudes,

but the discrete multitudes

required for

arithmetic can be

generated

from them by dividing a continuous

magnitude

into equal

parts. The object of arithmetic is therefore ultimately geometrical,

and

the

science

of

arithmetic is subordinate to geometry.

Hobbes expresses this opinion

in

his 1660 dialogue Examinatio

et emendatio mathematicae hodiernae, which contains

a

commentary

on Wallis's Mathesis

Universalis and aims

at nothing

less

than

the

wholesale refutation of Wallis's opinions.

In the

course of

the

dialogue, geometry

is defined as "the science by which we know

the

ratios between

magnitudes,"

or

more precisely

as

"the

science

of determining

magnitudes, either

of

bodies, or times, or

any other

non-measured

magnitude by comparing

it with another

measured

magnitude or magnitudes" [Examinatio, 1; OL, 4, 27]. Hobbes

then

remarks

that because "any

given continuous

magnitude

can

be

divided

into any number of equal parts,

with

its ratio

to

any

other magnitude remaining unchanged,

it

is manifest that

arithmetic

is contained in

geometry" [Examinatio,

1; OL,

4,

28].

He

later

adds that:

"Arithmetic is a

part

of

geometry, but it

is

not

a

great part

of it.

For from the purely geometrical

books

of

Euclid,

arithmetic can

easily

be derived. But the arithmetical

books,

even all that have been

written,

or

those

that Wallis will ever write, do not suffice to produce the

hundredth

part

of

the theorems

of

geometry which we

now

have."

[Examinat io,

3; OL, 4, 96.]

A

study

of Hobbes's

philosophy

of

mathematics thus

gives pride

of place to his philosophy of geometry, and I will be concerned

primarily

with his views

on

the

nature of geometry. Observe

also



Hobbes on the

Methods


that

in

claiming

arithmetic

to

be

a

special

case

of

geometry,

Hobbes

casts doubt

on analytic

methods. Because geometry is more general

than arithmetic,

there is

no

guarantee that

arithmetical

principles

will

apply

to geometry,

and

the

use

of

analytic techniques

must

therefore be

viewed with

suspicion.

1 /

Hobbes on Geometrical

Objects

We

have

seen

that

Hobbes's

materialism

leads

him

to

view

geometry

as a

science

whose objects

are

produced by

motion

and

extension, but

it

is not immediately clear what consequences this has.

This materialistic

program

emerges

most clearly in

Hobbes's

1666

work De principiis et

ratiocinatione

geometrarum

(18). In this essay,

Hobbes systematically critiques the

traditional

account of

geometric objects (drawn from

the

works

of Euclid

and

the

Jesuit

mathematician Christopher Clavius),

proposing

his

own

new foundations

for the geometric

science.

Hobbes is convinced a proper

understanding of

geometric

objects will

not

only

rid

geometry of false

principles

and confusion

but will

also

generate new

results

and

settle

disputes.

His plan is

therefore

to replace the

traditional

definitions

with

ones that treat

geometry

as a science of body.

Euclid's first definition is of a point,

and

reads: A point is

that which has

no

part" [Elements, I, Def. 1]. Hobbes objects,

claiming that the definition is

ambiguous. He notes

that we

can

interpret

the talk of a point having no parts

to

mean

either that

a point is indivisible or undivided. In the first sense, a point is

simply nothing as he

argues

in his

Six

Lessons to the Savilian

Professors:

"That which is indivisible is no

quantity;

and if

a

point

be

not

quantity, seeing

it

is

neither

substance nor quality,

it

is nothing. And if Euclid

had meant it so in his definition,

[...]

he might have defined it more

briefly,

but

ridiculously,

thus,

a point is nothing. [Six

Lessons,

1 ; EW,

7, 201.]

Defining a

point

as something undivided but capable of further

division

will

indeed make it a

quantity,

but the

definition

fails

(18)

See

[Sacksteder,

1981] for another

account

of

these matters,

based

on Part III

of

De

Cor

pore.



170

Douglas

M.

Jesseph

to express the essence of a

point. For

Hobbes, the essence of a

point is that its magnitude is not considered

in

a

demonstration,

and

he

defines

a

point

in

exactly

these terms:

"Therefore the

definition of

point in

Euclid, however

it is

understood

by all the geometers

after

him, is faulty... The true definition of

point, and one which

will bring

no

flaws

into demonstrations

must

be

of this sort:

a

point

is something divisible, but none of whose parts

are

to be considered in a

demonstration." [PRG,

1; OL,

4, 392.]

Hobbes must therefore claim that

points

are extended — a rather

shocking

departure

from the

traditional

conception of geometric

points.

Moreover,

the

definition would

seem to admit

points of

different

sizes, since

there

could

certainly

be

different

sizes of

divisible objects

whose parts are not

considered in

a demonstration (19).

The next geometric object we need to

consider

is the line. In

Euclid, a

line

is breadthless length" [Elements, I, Def. 2].

Hobbes

rejects

this for the

same reason

that

led him

to

dismiss the

Euclidean

definition of

a point: he holds that a

length without breadth

is

simply nothing.

He observes that other writers (notably

Clavius

in his commentary

on

Euclid's Elements) endorse a definition

of

the

line

in terms

of

the motion

of

a point (20), and he finds

such

definitions more

congenial. This should

be no

surprise

because

Hobbes'

attempt to

reduce

geometry to

motion and extension

would

be advanced by including the concept of motion

in

the definition

of

a line.

And

yet, he cannot follow Clavius and

define

a

line

as

the path

traced

by an

indivisible Euclidean

point,

for that

would

be to make the

line

produced by the

motion

of nothing.

Thus,

the

line can

be defined as the path traced by a moving point,

provided that the

term

point is understood in Hobbes's sense:

(19) Hobbes

explicitly

endorses this

unusual

view in his

discussion

of

conatus

in

De

Corpore:

And

yet,

as

a

point

may

be

compared

with

a

point,

so

one

endeavour

may

be compared with

another

endeavour, and one

may

be

found to

be

greater

or less than

another.

For

if

the

vertical points of two

angles be compared, they will

be equal

or

unequal

in the

same

proportion

which

the

angles

themselves have to one

another.

Or

if

a strait

line cut

many

circumferences

of concentric circles,

the inequality

of

the

points of

interse tion will be

in

the

same proportion which

the

perimeters

have to one another [DeC,

3.15.2;

EW, 1, 206-207]. Hobbes's doctrine of conatus and its relation to his

account

of

geometric

points

and infinity

are

treated

in

[Robinet, 1990].

(20)

Clavius

reports that In

order to teach the true understanding of a

line,

mathematicians

also

imagine

a point

as described in

the prior definition to be moved out of

one

place

and into

another.

Since the

point

is

wholly indivisible [prorsus individuum] it

will

leave behind from this imagined motion

a path having

length but

without any

breadth

[Clavius,

1612,

vol.

1,

13].



Hobbes on the

Methods


171

"[EJveryone

knows

that

nothing

except

body

can

be moved,

nor

can

motion be

conceived

of

anything

except body. And

every

body in motion

traces

a path with

not only

length,

but

also breadth. Therefore,

the

definition of

a

line should be as follows:

a

line is the path

traced

by

a

moving body,

whose

quantity is not considered in

a demonstration."

[PRG,

2; OL, 4,

393.]

This

definition

will

apply

both to

straight and curved lines, so

a special definition of straight (or right)

line

is needed. Euclid

defines a

right line

as "a

line which lies evenly

with the points

on itself" [Elements, I, Def. 4]. This definition is

famous

for its

obscurity

and

provided

commentators

the opportunity to

demonstrate their subtlety

in

unraveling it (21). Hobbes simply dismisses

it as useless and incomprehensible, and offers

an

alternative that

does not fall

prey

to the incoherence he finds

in

Euclid. This

definition

is in terms

of

motion, but not

simply

the motion

of

a point.

Hobbes defines a right

line as

a special case of a line,

namely

one

whose termini

cannot

be drawn apart without altering its

magnitude (22). Since the

quantity of

a line is the greatest distance

which

may be placed between its

endpoints,

Hobbes's definition of the

right

line

thus

implies

that it

is

the

shortest

line connecting two

points

(23).

The image

here

is that a straight

line

cannot have

its

termini

drawn farther

apart,

but

the end

points of a curved line

can

be separated while the

line

retains the same length. The

result

is that

we must

consider

two kinds of

motions in

defining a right

line: first the motion of a point (by which a

line

simpliciter is

traced), and then a motion drawing the termini

of

the

line

apart

from one

another.

If

the

second

kind of motion cannot be

conceived

without altering the

magnitude

of the line,

it

is a right line.

(21) For

a

recent account

of

the definition and

its

difficulties, see [Federspiel, 1991].

T. L.

Heath's commentary

in [Euclid, 1956,

1, 165-169]

is also quite

useful.

(22) "A

right line is one whose termini

cannot

be drawn apart

while

keeping the

same

quantity [PRG,

4;

OL,

4, 395].

The

same

definition appears

in

De Corpore: And

seeing

the

action,

by

which

a

strait line

is

made crooked,

or

contrarily

a crooked line

is made

strait,

is nothing but the

bringing

of its extreme points nearer to

one

another, or the setting

of them further

asunder,

a crooked

line

may rightly be defined to be that, whose extreme

points may be

understood

to be drawn

further

asunder;

and

a

strait

line to be

that,

whose

extreme points

cannot

be drawn further asunder; and comparatively, a more

crooked,

to be that line whose extreme points are nearer to one

another

than those

of

the

other,

supposing

both lines

to

be

of

equal

length

[DeC,

2.14.2;

EW,

1,

177].

(23)

For the

magnitude

of a

line

is computed by the

greatest

distance which may

be between its extreme points [DeC, 2.14.1; EW, 1,

154].



172 Douglas

M.

Jesseph

Given these definitions, Hobbes can define surfaces

and solids

easily in

terms of further

motions.

The Euclidean

definitions

are,

as

before,

unacceptable

to Hobbes. Euclid

declares

"a

surface

is

that

which has length

and breadth

only", while "a plane

surface

is a

surface which lies evenly

with the straight lines

on itself"

[Elements, I, Defs. 5, 7].

Clavius,

in

attempting to elucidate the

definition of "plane

surface" writes that

A

surface, then,

which lies evenly with the lines on it, so that the

middle parts

do not

deviate

from

the

extreme

parts by rising up or

going

down,

will be called

a

plane

surface,

such as the surface of some

highly

polished

marble, in

which all parts

are

arranged

in

a right

line,

so

that

they

do not form

angles, there

are

no bends,

nothing sticks

out,

and

there

are

no gaps."

[Clavius,

1612,

vol.

1,

15.]

Hobbes dismisses both Euclid and Clavius for

familiar

reasons:

"This matter

itself is well enough understood by both Euclid and

Clavius, and all other men. But not everyone can

express

in words, or

at least

not

easily, what is essential to a

plane

surface. Yet if you

say

that

a

plane surface

is

one which

is

described

by a line so

moved that

each

of

its

points describes

a

right

line,

then you will have defined it

well, and clearly, and agreeing with its essence." [PRG,

7;

OL, 4, 398.]

It

is

important

to

observe

that

Hobbes

does

not

define

a

surface

as

the

product

of two lines.

In

his view,

the

"drawing of

lines

into

lines" is a quintessentially geometric operation entirely

different from arithmetical multiplication. Although

the area of

a

rectangle can be computed by multiplying the

lengths

of its sides,

this is not the process by which the rectangle is created.

This

account of the

generation

of plane surfaces has important

consequences

for

Hobbes's

mathematical

enterprise.

Among other

things, it leads

him

to

mark an important difference between

arithmetical

and geometrical operations

and

to

hold

that

the

application

of algebra

or arithmetic to

geometry

is not guaranteed to

produce

true

results.

Hobbes

stresses

the

difference between

geometry

and

arithmetic

with

reference to the

genesis

of geometric

magnitudes when he declares:

A

square figure

is made by

the

drawing of a right line

into an

adjacent

equal line at right

angles.

This drawing

therefore will describe

a surface, that is, a new species of quantity: and

this

surface drawn

into

another

right

line equal

to the first two and orthogonal to them

[in aliam

rectam aequalem prioribus

et

erectam] will describe a solid,



Hobbes on the Methods of Modern Mathematics

173

that

is,

the

third

and

last

species

of quantity.

For

geometry

knows no

quantity beyond the solid. The 'sur-solids', 'quadrato-quadrata',

'quadrato-cubes', and 'cube-cubes' of the

arithmeticians

are

mere

numbers

and

not

figures.

Therefore

there is a

great difference

between

that

which

is done by the drawing of two lines into one another and that which

is done by

the

multiplication of

two

numbers: clearly

as great

a

difference as that

between a line

and

a surface,

or

between a

surface and

a

solid. These differ in species, so that one

can

never exceed the other

by any multiplication, and

thus,

as

Euclid

said, they

can

have no ratio,

nor

can

they be compared to one another according to quantity. There

is likewise

a

great difference

between

the root of

a

square

number

and

the

side of

a

square

figure.

For

a

root

is

a

number

and

an

aliquot

part

of

its

square, but the side of

a

square figure is not

a

part of

it."

[LM,

Preface; OL, 5, 96-97.]

The

insistence here upon the "great

difference" between

arithmetical roots and geometric lines will reappear when we

consider

Hobbes' s

rejection

of

analytic methods and his dismissal

of

algebraic

refutations of his

putative

results. We

need

simply

note

here

that Hobbes's insistence on this doctrine contributed to the

disastrous failure

of

his

controversy

with Wallis, for it

led

him to

repudiate

nearly

all

of

classical

geometry.

Once

the simple

figures

such

as

squares

and

rectangles have

been

defined,

complex plane figures

can

be defined

in

the obvious

way

by considering

non-rectilinear

motion

of

lines or motions in

which the

line

is diminished or increased. A

figure

of special

interest is the

circle, which Euclid

defines as "a

plane

figure

contained by one line such

that

all the straight

lines

falling upon

it

from one

point

among

those

lying

within

the

figure

are equal to

one another"

[Elements, I,

Def.

15].

Hobbes admits that

this

definition gives a true description

of the circle,

but

faults

it for

failing

to

give a

causal

account of its

generation.

In the first of

his Six

Lessons he argues:

"But

if a man had never seen

the

generation of a circle by

the

motion

of

a

compass

or other equivalent means, it would have been hard to

persuade him

that

there was

any

such figure possible. It

had been

therefore not

amiss

to have

let

him

see

that such

a

figure might be

described.

Therefore so much of geometry is no part of philosophy, which

seeketh

the proper passions

of all

things in the

generation of

the things

themselves. [Six

Lessons,

1; EW, 7,

205.]

In

Hobbes's treatment the circle must

be defined as

the

figure traced by the rotation of a

line

about

one

of its ter-



174

Douglas M.

Jesseph

mini (24). At work here

is his

fundamental methodological

principle that a

genuine demonstration

must proceed from causes.

It

is

worth noting here

that

Hobbes

strives

for

causal

definitions"

in his

treatment of the

foundations

of geometry

for

two reasons:

first because

he

holds

that true scientific

knowledge must be

knowledge of

causes,

and

second

because he insists that all science must

begin with

proper definitions (25). These

considerations result in

Hobbes' s

confidence that he had uncovered the means

by which

to solve all geometrical problems.

Previous

generations

of

mathematicians had failed

to

find

the

true quadrature

of

the

circle,

but

(at

least

on Hobbes's

understanding

of

the

matter)

this failure did

not stem an intrinsic intractability of the problem or from a lack

of

industry

and

intelligence

on

the

mathematicians

part.

Instead,

their efforts

were

in vain because

they began from obscure

principles that failed to disclose the true causes of geometric figures.

But

once the true

causes

have been

brought

to light,

there

is

no

deep mystery

about

how to proceed in finding the sought result.

2 / Analysis, Synthesis

and

Mathematical

Demonstration

The

prominence of

causal

principles

in

Hobbes's

philosophy

of mathematics has several consequences. In the first

place,

it leads

him to

classify

mathematics as a branch

of

philosophy. Hobbes

defines

philosophy as

"such knowledge of

effects or

appearances,

as

we

acquire

by true ratiocination

from

the knowledge we

have

first

of

their

causes

or generation:

And

again,

of

such

causes

or

generations as

may be from

knowing

first

their effects"

[DeC,

1.1.2; EW, 1,3]. The mathematical

philosopher understands

the

(24)

In

De

Corpore

he

defines

the

circle

as

follows:

"If

a

strait

line

be

moved

in

a

plane, in such

manner,

that

while

one end

of

it

stands

still, the whole line be carried

round about till it come again into the

same

place from

whence

it was

first

moved, it

will describe

a

plane superficies,

which

will be terminated every way by that crooked line,

which

is made by that

end

of the strait

line

which

was carried round. Now this superficies

is called a CIRCLE" [DeC,

2.14.4;

EW,

1,

180-181].

(25)

Thus Hobbes defends his appeal to motion in the definition of figures with the

argument

that All

demonstrations

are

flawed,

unless they are scientific; and unless they

proceed

from

causes, they are not scientific. Secondly, they are flawed unless their

conclusions are demonstrated by construction, that

is,

by the description of figures, that is

from

the drawing of lines. For every drawing of a line is a

motion.

And so

all

demonstrations

are flawed,

whose first principles

are

not

contained

in the definitions

of

the motions by

which

figures

are

described

[PRG,

12; OL,

4,

121].



Hobbes

on

the Methods of Modern

Mathematics

175

causes

of geometric

objects

by

grasping

the

appropriate

definitions,

and his understanding of these

causes

yields knowledge

of the effects,

namely the

geometric objects

themselves. Indeed, it is

the

accessibility of causes

that

makes geometric

demonstration

possible. Hobbes

makes this clear in

the

dedication

to

his

Six

Lessons to the Savilian

Professors of

Mathematics,

when

he links the

demonstrability of

mathematics

to the fact that we are the cause of geometric objects:

"Of arts, some

are

demonstrable, others indemonstrable; and

demonstrable are those the construction of the

subject

whereof is in the power of

the artist

himself,

who in his demonstrations

does

no more

but

deduce the

consequences

of

his

own

operation.

The

reason

whereof

is

this,

that the

science of every subject is derived from a precognition of the causes,

generation,

and construction of the same; and consequently

where

the causes

are known, there is

place for

demonstration,

but not

where the

causes

are

to seek for. Geometry

therefore

is demonstrable,

for the

lines and figures

from

which we reason are drawn and described by ourselves and

civil

philosophy is demonstrable,

because

we make the commonwealth ourselves.

[Six Lessons, Preface;

EW,

7,

183-184.]

As noted earlier, this

doctrine explains much about

Hobbes's

confidence in his

ability

to solve

such

notorious problems as the

quadrature

of

the

circle.

He

imagined

that

he

had hit upon

the

true,

causal

account of

geometry;

and his

general methodology

dictated that

anyone fortunate enough to know

the cause of something can

(at

least

in principle) obtain

complete knowledge of it.

Thus,

Hobbes holds

that all properties of the circle should

become manifest

once the true

account of its genesis has

been grasped.

Consideration of the order of

causes

and effects allows Hobbes

to introduce a fundamental

distinction

between

analytic

and synthetic

reasoning.

This

celebrated

part

of Hobbes's

system

has

been

the

object

of much scholarly

discussion

over

the

years,

and

it

is

an

essential

part

of

his

treatment of

mathematics

(26).

Although

the account contains

complexities,

the basic idea is quite

simple:

synthesis proceeds from

causes

to effects, while

analysis moves

from effects to causes.

Mathematicians well

before

Hobbes had made a distinction

of

this kind,

(26)

See [Prins, 1990], [Sacksteder, 1980], [Talaska,

1988],

and [Watkins, 1965] for

discussions

of

Hobbes's conception

of

analytic and synthetic methods and their

relation

to science and mathematics. Watkins holds that Hobbes is indebted to theories

of

method

in the School

of

Padua in the

late

sixteenth-century,

particularly

in the

work of

Jacopo

Zabarella. Prins

disputes

this claim and argues

that

Hobbes and Zabarella have entirely

different

conceptions

of

science. Fortunately,

there

is

no

need

to

resolve

this

issue here.

As

we

will see,

at

least in the mathematical case, the distinction

between

analysis and

synthesis is

a commonplace

with

a long

history

before

Hobbes.



176

Douglas

M.

Jesseph

although

without

couching it in terms

of

causal

order. Pappus,

for

example, says that:

"

[A]nalysis

is

the

path

from

what

one

is

seeking,

as if

it

were

established,

by way of

its

consequences, to something that is established by synthesis.

That

is to say, in analysis we

assume

what is sought as if it has been

achieved,

and

look for

the thing from which

it

follows, and

again

what comes before

that,

until by

regressing in this

way

we

come upon some one principle...

In

synthesis,

by reversal, we assume what

was

obtained

last

in the

analysis

to have

been achieved

already, and, setting now

in

natural

order, as

precedents,

what before were

following, and fitting them

to

each other, we

attain

the end of the construction of what was sought. [Pappus, 1986, vol.

1,

83.]

François

Viète's

comments are

representative

of the seventeenth-century

understanding

of the

distinction between analysis and synthesis:

"There

is a certain way of

searching for truth in

mathematics,

which

Plato is said first to have discovered, and which

Theon

called analysis.

He defined it as assuming that which is sought as if it were admitted

and

proceeding through the

consequences

of that assumption to what

is admitted as true. This is opposed to

synthesis,

which is assuming what

is

already

admitted and proceeding through its consequences to arrive

at

and to understand what is sought." [Viète, 1646, 1.]

It was

also

traditional

to

distinguish

these two

methods

by

calling

synthesis

the "method of

demonstration" and

analysis the "method

of discovery",

with

the idea

that analysis is

a preliminary

to

synthesis. On this

way of

looking

at

the matter analysis provides

a means of exploring the

conditions

under which a problem can

be solved or a theorem

proved,

and

it

can thus be used to uncover

the fundamental principles that

suffice

for a true synthetic

demonstration of

a proposition. If

analytic

reasoning leads back to primary

truths

such

as

axioms,

and

if the steps

in

the process are

convertible (in the

sense

that

there

is a logical

entailment

from the

consequent to the antecedent), then

an

analytic procedure

can

be

turned

into a

synthetic demonstration

from the axioms.

Although Hobbes sanctions both analytic and synthetic

reasoning

in

geometry,

he

holds

that

only

synthesis

can be truly

demonstrative.

The reason for this should be clear:

analysis

proceeds

hypothetically,

but synthesis

leads

from acknowledged first

principles

to their necessary

consequences.

Synthetic reasoning thus

satisfies the

traditional

requirement

that

demonstrations proceed

from

principles

better known and more secure than their conse-



Hobbes

on the

Methods


177

quences,

and

it

satisfies

Hobbes's

requirement

that

true

knowledge

be

grounded in

the investigation of

causes. Hobbes

thus

holds that

analytic reasoning

is acceptable

mathematics

only

in so

far

as it

can set the

stage for a demonstrative

synthesis,

and it is in this

sense that he claims

no man can even be a

good

analyst without

being first a good geometrician;

nor

do the rules of analysis make

a good geometrician, as synthesis

doth; which begins at

the very

elements and

proceeds

by a logical

use

of the same. For

the true

teaching of

geometry

is by

Synthesis,

according to Euclid's method"

[DeC, 3.20.6; EW, 1, 314.].

III.

—

Hobbes

and Analytic Geometry

Hobbes is notorious for his rejection

of

analytic

geometry.

Indeed,

some of

his more

memorable polemics are directed

against

the

use

of

algebraic and

analytic

techniques in

the solution of

geometric

problems.

Thus, he dismisses Wallis's Treatise of Conic

Sections

as

a work

so covered

over

with

a scab of

symbols,

that

I had not the patience to examine whether it bee well or ill

demonstrated

[Six Lessons,

5;

EW, 7,

316].

In one particularly

unrestrained

outburst,

he

critiqued Wallis by

asking:

"When

did

you

see any man but yourself

publish

his

demonstrations

by

signs not

generally received, except

it

were

not

with intention to

demonstrate, but to

teach

the use of

signs?

Had Pappus no analytics?

or

wanted

he the wit

to

shorten

his reckoning by signs? Or has

he not

proceeded

analytically

in

a hundred problems (especially

in his

seventh

book),

and

never

used symbols? Symbols

are

poor unhandsome,

though

necessary

scaffolds of

demonstration; and

ought

no

more to

appear

in

public, than the most deformed necessary business which you

do

in your

chambers."

[Six Lessons,

3; EW,

7, 248.]

This much

of

Hobbes's case

against

algebraic methods is

hardly worth

taking

seriously, since

it

amounts only to an "aesthetic" complaint

that

symbolic methods deface geometric

demonstrations.

But

he

adds

a more interesting criticism when he contends that

the

introduction

of algebra

cannot

add

anything

to geometry

because it

distracts the

geometer's attention from geometric

magnitudes

and replaces the

contemplation

of magnitudes

with

the

manipulation

of symbols.

This

is

the

import

of

his

accusation

that

Wallis

is

misled by empty

symbols:




I verily

believe

that

since

the

beginning

of the world,

there

has not

been, nor

ever

shall be, so much absurdity written

in geometry,

as is

to be found in those

books

of [Wallis]... The cause whereof I imagine

to

be

this,

that

he

mistook

the

study

of

symbols

for

the study

of

geometry,

and thought

symbolical

writing to be

a new

kind of method, and

other men's

demonstrations

set down in symbols

new

demonstrations.

The way of analysis by

squares, cubes &c,

is

very ancient, and

useful

for

the finding out whatsoever is

contained

in the nature and generation

of rectangled planes, which also may be found without it, and was at

the highest in Vieta; but I never

saw

anything added

thereby

to the science

of

geometry,

as being a way wherein

men

go round from the equality

of rectangled planes to the

equality

of proportion, and

thence

again to

the equality of

rectangled

planes, wherein the symbols serve only to

make

men

go

faster about, as greater

wind to a windmill. " [Six Lessons, Epistle;

EW,

7,

187-188.]

Where others had taken algebra to be a

way

of

simplifying

demonstrations while also enabling geometers to find

new

results, Hobbes

sees algebra

as

only

a

new

kind of language that has been foisted

upon

geometry

to no

purpose.

In a similar

vein,

he

claims

that

algebra

cannot shorten demonstrations or make geometry easier

to understand.

Contrary

to the

claims

of

the modern

analysts,

Hobbes insists that "algebra

can

yield brevity

in

the

writing

of

a demonstration, but not brevity

of thought.

Because it is not

the

bare

characters,

or

only

the

words,

but

the things

themselves

which are

the

objects of

thought, and these cannot be abbreviated"

[Examinatio, 3; OL, 4, 97]. As Hobbes sees the

matter,

a proper

demonstration

must

proceed

by way of

constructions from

causes

to

results, but

reliance

upon algebra simply interposes a collection

of

symbols between ourselves and the magnitudes we

are

to

construct.

Hobbes's

distinction between

analytic

and synthetic reasoning

also accounts, in part, for his rejection of analytic geometry. A

typical

proof

in

analytic geometry

proceeds

"analytically"

in

the

sense outlined

above, by

first

supposing

the

problem

solved and

then showing that

the solution

is

algebraically admissible. As

Descartes puts it:

"Ainsi voulant résoudre quelque problesme, on

doit

d'abord le

considérer

comme

desia

fait,

& donner des noms a toutes les lignes,

qui

semblent nécessaires pour

le

construire, aussy bien

a celles

qui sont

inconnues, qu'aux autres. Puis sans considérer aucune difference entre

ces

lignes

connues, & inconnues, on doit par courir la

difficulté,

selon l'ordre qui

monstre

le

plus

naturellement de tous en qu'elle

sorte

elles

dependent



Hobbes

on


Mathematics

179

mutuellement

les

unes

des

autres,

jusques a

ce qu'on

ait

trouvé moyen

d exprimer

une

mesme quantité en deux

façons

: ce qui se nomme

une

Equation; car les terme de

l'une

de ces deux

façons

sont esgaux

a

ceux

de l'autre. Et

on doit trouver

autant de telles

Equations, qu'on

a

supposé

de lignes, qui estoient inconnues." [AT, 6, 372-373.]

But this haphazard approach

is

not a proper

demonstration

by

causes and,

Hobbes claims,

can at best result

in a half-

finished

demonstration that must still be "converted" into a synthesis. True

geometry must appeal to causes and

constructions,

rather than the

hypothetical

procedure

of

Descartes

and

its

fortuitous unravelling

of

a problem

by

means

of

equations.

We can thus discern two strands

in

Hobbes's objection to

analytic geometry: he opposes

both

its excessive

use

of algebraic

symbols and its

reliance

upon

hypothetical

procedures. These two

strands

of Hobbes's critique of analytic

geometry can

be drawn

together

and

phrased

in a single

objection:

the use

of

algebraic

methods in

geometry must be

either

unscientific

or

superfluous.

For if algebraic

techniques

have governing principles and

do

not

simply proceed

par hasard, then

these

principles

must be

vindicated by appeal to geometric considerations, thereby making

algebra

superfluous. Hobbes poses this dilemma

in

the Examinatio as

follows:

"What else

do the great

masters of

the current

symbolics, Oughtred

and Descartes, teach, but that

for

a

sought quantity we should

take

some

letter

from the alphabet, and

then by right reasoning

we

should proceed

to the consequence? But if this be an art, it would need to have been

shown what this right reasoning

is.

Because

they do not do

this,

the

algebrists are known to begin sometimes

with

one supposition,

sometimes

with

another,

and

to

follow sometimes

one

path,

and

sometimes

another...

Moreover,

what

proposition discovered by algebra

does not

depend upon Euclid (VI, 16) and (I, 47), and other famous propositions,

which one

must

first know before he

can

use the rules of

algebra?

Certainly, algebra needs geometry, but geometry

does

not need algebra."

[Examinatio,

1;

OL, 4, 9-10.]

Clearly,

the

right"

reasoning

to which Hobbes refers

in this

passage

will be

reasoning from

causes to

effects, that is to

say

demonstrative

knowledge grounded in a synthetic exposition

of

the

properties of geometric objects.

In

Hobbes's view, we

cannot know

whether

an

algebraic procedure

is

legitimate

unless we already know

that

the geometrical step which

corresponds

to

it is

an admissible




construction.

But

in

such a case

there

is no need to distract

ourselves with

the study of algebraic

symbols, and

we should

proceed

immediately

to the

construction.

Hobbes took his case

against analytic methods

to the

extreme

of declaring that the

application

of

algebra

to geometry produces

false

or irrelevant

results.

Consider, for

example,

some

of

his

remarks

in

reply

to Wallis's

refutation of a

purported

solution

of

the

Delian

problem"

of

doubling the

cube. Wallis

had shown

by

algebraic calculation that

Hobbes' "solution" was in

error

(27).

Hobbes replied in

an appendix

to his Problemata Physica,

insisting:

In the

examination

of this

calculation,

the

difference

between

the

multiplication

of the arithmeticians and the "drawing of lines

into

lines"

of

the

geometers

is

to

be noted,

quite

in itself

and

without considering

demonstration

in mathematics" [Duplicatio

Cubi;

OL,

4, 380]. As we saw earlier, Hobbes

claims

that when

lines

are drawn into

one another to

produce a surface

(or

when

a surface is drawn into a line to create a solid), a geometric object

of a

higher

dimension is

produced

from two objects of a lower

dimension.

This "change

in species

of magnitude" is not found

in

arithmetical

multiplication,

where

numbers

are multiplied

together

to obtain numbers, but never objects of a

different

type.

Moreover,

geometry

is

confined

to

three

dimensions

where

algebra

and arithmetic

can ascend

to fourth and higher powers

by

continued multiplication.

Thus,

while geometric squares

and cubes

are

truly geometric objects, the so-called

squares

and cubes of algebra

and arithmetic

are simply

numbers.

In

examining Wallis's

algebraic

refutation of

his

solution,

Hobbes

argues

that Wallis

sometimes treats

a

unit as

a line, sometimes

as a square, and sometimes as a cube; the

result

of this procedure

is to

misrepresent the

geometric situation

by

applying

algebra

to

it.

A

given number

appears

first

as

a

line,

then

a

square,

then

a cube, and then

again

a square root. Hobbes concludes that

the

problem with algebraic methods is that they misapply the notion

of a

unit

(a concept restricted to arithmetic) to geometric objects

which

do not consist

of collections of

units:

"Therefore

the

calculation in numbers where

any

line is

taken

for unity is

necessarily false

(27) Hobbes's

argument

fails by requiring

(3

- V2)

=

45 -

V1682, which

amounts

to

ignoring

the difference

between

VÏ68Ï and

V1682.

Wallis refutes Hobbes's alleged

demonstration in [Wallis, 1669].



Hobbes

on


Mathematics

181

(as

a

line

is

always

divisible

into

divisibles,

but

unity

is

not

divisible), and hence it is

of no

use for confirming or refuting a

geometric

calculation" [Duplicatio Cubi;

OL,

4, 384].

These remarks

show just how

widely Hobbes's conception

of geometry ultimately

deviates

from the

traditional

understanding of the subject. In defense of his failed

claims

to

mathematical glory, Hobbes is

forced

to deny that arithmetic or

algebra

have any

relation

at all

to

geometry,

and finally to

insist

that

the extraction of roots as

practiced

by Euclid, Archimedes, and

all

other

geometers

is based upon

a

colossal

mistake. In

fairness

to Hobbes, we should note that the

use

of algebraic

methods

in geometry did pose

important

conceptual problems, particularly

for mathematicians of the

seventeenth

century. These problems

arose because it was not always clear how the algebraic

operations

were to be interpreted

in

geometry or what

principles licensed

particular

algebraic moves

in any given

construction.

But these

problems were solved through a reinterpretation of the relationship

between

geometric constructions

and algebraic operations and not

(as Hobbes would

have

it) through the elimination of

algebra.

In the

end,

it seems that

Hobbes's

fundamental problem with

the

new geometry

stems from his refusal to grant Descartes' s

fundamental

assumption of the unity of geometric and algebraic

magnitudes.

Descartes's

program requires

a

reinterpretation

of

the

notion

of geometric

multiplication and other operations in

which all geometric

operations

be

confined to

line segments

and

the

computation

of relations

among

them.

Hobbes

simply refused

to

accept

this

way of

proceeding

and

clung

to an

older

conception of geometry.

IV.

—

Hobbes

and the

Method of

Indivisibles

Hobbes's wholesale rejection of

analytic methods

contrasts

with his much more

complex

and ambivalent

attitude

toward

the

method

of

indivisibles. Although

some

commentators

have

portrayed Hobbes as hostile to the

method

of indivisibles tout



182 Douglas

M.

Jesseph

court, this is not a strictly accurate

account

of the matter (28).

Hobbes was tireless and

uncompromising

in his criticisms

of Wallis's

use of

the

method,

but

he

accepted Cavalieri's

presentation of

it.

Indeed, Hobbes' s

own mathematical

work

in

De Corpore is drawn

almost directly from Cavalieri's Exercitationes

Geometricae Sex.

The aim of this

section

is to document this difference

in

Hobbes's

attitude

toward the method

of

indivisibles and to explain how he

could endorse the method as practiced

by Cavalieri while

vilifying

Wallis's

employment of it.

Hobbes made no great secret of his favorable attitude toward

Cavalieri.

In the

Admonitio

ad

Lectores" at

the end

of Lux

Mathematica

he

declared:

"We had, avid reader,

very skillful masters

of

the

human sciences

(I speak of geometry and physics) in the most distant ages: above all

in geometry Euclid, Archimedes,

Apollonius,

Pappus, and others from

ancient Greece. More recently we have

Cavalieri

and Torricelli from Italy...

But today, I

say we

are

not even

staying even,

but

instead

are

falling

ackward." [LM,

14;

OL, 5,

147-148.]

Both

Cavalieri

and Torricelli acquired their mathematical

reputations

through an

exploitation

of

the method

of

indivisibles, so this

praise

for

the

two strongly

suggests

that Hobbes held

the

method

in

reasonably high regard, especially

in

contrast to

later

procedures which he took to be a decline

in

mathematical standards.

Elsewhere, Hobbes continues this

line of thought

when he insists

that Wallis's conception of

points

as unextended

"destroys

the

method

of indivisibles, invented by Bonaventura; and upon which,

not well

understood,

you have

grounded

all

your

scurvy book of

Arithmetica

Infinitorum" [Six Lessons, 5; EW, 7,

301].

Again,

this comment clearly indicates that Hobbes approved

of

Cavalieri's

work.

Even

stronger

evidence

for

Hobbes's

acceptance

of

the

method

(28) [Robinet, 1990,

148-149]

holds that

Hobbes

déplore

que

l'agrégat

des indivisibles

de Cavalieri ne puisse

jamais être

égal

à une grandeur

donnée. He

adds: Or

l'indivisible

est loin d'être la partie que recherche Hobbes. En effet la définition de la géométrie des

indivisibles tombe sous le coup d'une triple objection. The

three-fold objection

that Robinet

has in mind here concerns the foundational

problems

surrounding the composition of

continuous

magnitudes

out

of indivisibles, but

he

fails to

observe

that these

polemics

are

directed against Wallis and not

Cavalieri. In

a similar fashion, [Mancosu and

Valiati,

1991]

take Hobbes's critique of Wallis

as

a rejection of the

infinite in

mathematics, and

presumably Cavalieri's conception of indivisibles

as well.



Hobbes

on


Mathematics

183

of indivisibles

comes

from the

manuscript collection

at

Chatsworth.

Among his surviving papers is a notebook in

which

Hobbes

copied

extracts

and summaries

of

Cavalieri's Exercitationes

Geometricae

Sex (29).

The

notebook was

used

by Hobbes during his extended

stay

in France in

the

1640s — the period in

which

he

assembled

his De Corpore

and

when Cavendish enlisted his aid

in Pell's

campaign against Longomontanus (30).

De

Corpore

itself contains the

strongest evidence of

Cavalieri's

influence

on Hobbes. The ill-fated circle quadrature

in Part III,

Chapter 20 of

De

Corpore

follows a

long

account

of motion,

magnitude, figure, and measure which betrays a significant debt

to

Cavalieri's

ideas. In particular, Chapter 17 of De Corpore

on

the measure

of "deficient

figures" comes

almost straight

out

of

the

Exercitationes

Geometricae Sex,

as we

can see by

comparing

its second

article with

proposition 23 of part four of

Cavalieri's

Exercitationes.

In

Hobbes's

parlance, the deficient figure ABEFC in figure 4

is

produced by

the motion

of the line AB through AC, while AB

diminishes to a point at

C. The "complete

figure" corresponding

to

the deficient figure is the

rectangle

ABDC,

produced

by the

motion

of

AB

through

AC

without diminishing.

The

complement

of the deficient figure is

BDCFE,

the figure which, added

to

the

deficient

figure,

makes

the

complete figure. Hobbes's

task

is to

find

the ratio of

the

area of the deficient

figure to

its

complement,

given

a specified rate of decrease of the quantity AB. He concludes

that the ratio of the

deficient

figure to its complement

is

the same

as the

ratio

between corresponding

lines

in

the deficient

figure

and

their

counterparts

in the complement.

His

statement

of

the

theorem

reads:

(29) Chatsworth Collection,

Hobbes Ms.

C.I.5.

(30) The dating

of

the manuscript is not certain, but the hand and

subject

matter put

it in the same period as Chatsworth

Ms.

A.4,

a

draft

of

Chapter 19

of

De Corpore dealing

with mathematical

matters.

This dating

implies that

Hobbes copied

out

the extracts from

Cavalieri while he was writing De

Corpore.

The Exercitationes

were

published in

1647,

and both Hobbes and Sir Charles Cavendish must have

sought

it eagerly. In

a letter

to

Pell from 2 August, 1648, Cavendish wrote that

Mr:

Hobbes hath nowe leisure to studie

and I hope

we shall

have his within

a

twelve-month. He then

adds:

"I

saw a booke

at Paris of the excellent

Cavalieros

lately printed, concerning

Indivisibles

; whom you know

was

the Inventor or

Restorer of

that

kinde of

Geometrie ; I had no time

to

reade

it before

I came awaye, and they

are

not

to

be

bought;

Mr: Careavin comming latelie from Italie

brought

this with

him

[British

Library,

Add.

Ms.

4278,

f.

273r].

It

is

therefore

no

great

stretch

of

the imagination to think that Hobbes was

reading

Cavalieri as he put

together

the mathematical sections

of

De Corpore in 1648 and

1649.



184

Douglas M.

Jesseph

С

D

ig.

3.

— Based

on T.

Hobbes,

De Corpore, op. cit.

A

deficient figure, which

is made by a

quantity

continually decreasing

to nothing by ratios everywhere proportional and commensurable, is to

its

complement,

as the

proportion of

the

whole altitude to

an

altitude

diminished

in any

time is to

the

proportion of

the

whole quantity,

which

describes

the

figure, to

the

same

quantity

diminished

in

the

same

time."

[DeC, 3.17.2; EW, 1,

247.]

Thus, if

the rate

of

diminution

of

AB is

uniform,

the

line ABEFC

will be a right line (indeed, the

diagonal

of the

rectangle), and

the deficient

figure

will be to its complement

as

one to one.

In

more

complex

cases, as when AB decreases as the

square

of the

diminished

altitude, the area of the

deficient figure

will

be twice

that

of its

complement.

And,

in

general, if the line AB decreases

as the power n, the

ratio

of the deficient figure to its

complement

will

be /i:l.

Hobbes's

proof procedure for this

theorem

involves

the consideration of ratios between all the lines"

in

the deficient

figure and

its

complement, and

is nearly

identical with

Cavalieri's

famous procedure in

the

fourth

of

his six Exercitationes

Geome-

tricae

—

an exercise entitled De Usu Indivisibilium in Potestatibus

Cossicis.

There,

Cavalieri

pursued a result

which historians of

mathematics

generally

characterize as

the

attempt to prove the

geometric

equivalent

of

the

theorem

that the integral

of

the

function

x" on



Hobbes

on


Mathematics

185

the

interval

[0,

a] is

a{n+1)/(n

+

1).

Except for

differences

in

diagrams

and

terminology,

Cavalieri's

fourth

Exercitatio is the same

as

Hobbes's account of

deficient figures. In Proposition 23

of

Exercitatio

4, Cavalieri asserts his version of the

theorem

we saw earlier

from De

Corpore:

Fig.

4.

—

From B. Cavalieri, Exercitationes Geometricae

Sex,

op. cit., p. 303.

"In

any

parallelogram

such

as

BD

[as

in

Figure

4],

with

the base

CD

as régula, if any parallel to

CD

such as EF is

taken,

and if the

diameter AC is drawn, which cuts the line EF in G,

then

as DA is to

AF, so CD or EF will be to FG. And let AC be called the first

diagonal.

And again as

DA2

is to AF2, let EF be to FH, and

let

this be

understood

in all the parallels to CD so that all of these

homologous

lines

HF

terminate in the curve AHC.

Similarly,

as

DA3 is to

AF ,

let also

EF

be

to

FI,

and likewise

in

the

remaining parallels,

to describe the

curve CIA.

And

as AD4

is to AF4, let

EF

be to

FL, and likewise in

the

remaining

parallels to describe the curve CLA.

Which procedure

can

be supposed continued

in

the other

cases.

Then СНА is called the

second

diagonal,

CIA

the third

diagonal,

СНА

the fourth

diagonal, and so

forth.

Similarly the triangle AGCD is called the first

diagonal

space of the

parallelogram, the

trilinear figure

AHCD

is

the second

space,

AICD the

third,

ALCD the fourth, and so on. I say

therefore

that the

parallelogram BD

is

twice the

first space,

triple the second

space,

quadruple the

third space, quintuple the fourth space, and so forth." [Cavalieri, 1647,

279.1

A

consideration

of

the details

of

Cavalieri's proof

would

take us

too far afield, but

it is

enough

to

indicate

that

his

approach

was

clearly the model that Hobbes used

in

his treatment of deficient

figures.

The

similarity

between Hobbes and Cavalieri on

this point is



186

Douglas

M. Jesseph

certainly

striking, and it was

obvious to

Hobbes's contemporaries

that this approach to quadratures

had its

roots in

Cavalieri's work.

In fact, Wallis accused Hobbes of plagiarizing from

Cavalieri.

In

his

Due

Correction for Mr. Hobbes, or Schoole Discipline for not

saying

his

Lessons Aright

Wallis

levied the charge

by

printing

an

excerpt from a letter

alleging that

"those

propositions

which

Mr. Hobs had concerning the measure of the parabolasters were

not his

own,

but borrowed from somebody

else

without

acknowledging his author" and that

"they

were to be found

demonstrated in

an

exercitation

of Cavalerius

De

Usu

Indivisibilium

in

Potestatibus Cossicis'y

[Wallis, 1656, 7]

(31).

Aside

from the obvious

similarities

in

their methods,

there

are

other

important points of

similarity

between

Hobbes's

and

Cavalieri's approach to indivisibles. Both

conceive of indivisibles

as

described

by the

motion

of a line through a

figure, and

both avoid

speaking

in

terms of an infinite number of indivisibles

composing the relevant figures.

More

significantly, they

both rely

upon

the

calculation

of ratios

between

lines which is rooted

in

the

Euclidean

theory of

proportions rather than the algebraic treatment

of

magnitudes characteristic of analytic geometry. In contrast, Wallis

does

not appeal

to motion,

makes no

scruple

of speaking of

continuous

magnitudes

as

infinite

sums of

infinitely

small

elements,

relies

heavily

upon

arithmetical and algebraic methods, and does not

use the painstaking

comparison of ratios we

find in

Hobbes

and

Cavalieri.

Despite

his

departure

from Cavalieri's treatment of

indivisibles, Wallis

portrayed

his

own

work

as a simple continuation

of

Cavalieri's

work.

In so doing he

treated the foundational

questions

regarding the

method

as having been

resolved

by

Cavalieri

and

attributed to

him the view that

any continuous

magnitude is

composed

of an infinite

number

of indivisibles. Thus,

in

his Mecha-

nica,

sive

Tractatus

geometricus

de Motu, Wallis declares in

a

definition that "any continuum

(according

to the Geometry of

Indivisibles of Cavalieri)

consists

of an infinite number of

indivisibles

[Wallis,

1693-1699,

1,

645].

Wallis

illustrated

this doctrine

(31) The

author

of

the letter is known to us only as

a

British gentleman by the name

of

Vaughn. In

a

letter from Henry

Stubbe

to Hobbes dated 19 December, 1655 [British

Library, Add.

Ms.

32553,

f. 21], Stubbe reports from a Mr. Vaughn, whose brother's letter

was reprinted by Wallis without

permission. If

we

are

to believe Stubbe, Vaughn

did

not

intend his letter to be made

public

and was embarrassed by the episode.



Hobbes

on


Mathematics

187

by

claiming

that,

on

Cavalieri's

principles,

a

line

is

to

be

regarded

as an infinity of points, a surface as an infinity of lines, or time

as

an

infinity

of instants. Hobbes was sensitive to

this

misrepresentation of

Cavalieri

and charged Wallis

with

making a hash of

an otherwise

useful method. In

the Lux Mathematica

Hobbes

claims

that Wallis

... supposes two principles: the first is one

which,

so he says, comes

from Cavalieri, namely this: that any continuous quantity consists of

an infinite number of indivisibles, or of infinitely small parts. Although

I, having

read

Cavalieri's

book, remember

nothing

of this

opinion in

it,

neither

in the

axioms,

nor

the

definitions, nor

the

propositions.

For

it is false. A

continuous

quantity is by

its

nature always divisible into

divisible

parts: nor can

there be

anything infinitely

small,

unless

there

were given a division into

nothing."

[LM, 3; OL,

5,

109.]

Hobbes's

hostility

to Wallis's method of indivisibles is not,

however,

confined to the claim that Wallis misinterpreted

Cavalieri.

In

numerous

passages

he

notes

that Wallis's

conception of surfaces

as

composed of

lines

leads to apparent paradox (32). If,

indeed,

a line

is "breadthless length", then the

addition of

numerous lines

could

never

constitute

a

breadth

because

it

would

simply

be

a

sum of

the form 0

+

0

+

0...

Hobbes concludes that

Wallis's approach

to

the method of

indivisibles

is

simply incoherent, since

indivisibles

appear to be

both

something and nothing,

while

surfaces are

somehow composed

out

of collections of "nothings" (33). Similarly, by basing his

method

on the

computation

of ratios between infinite

sums

or series, Wallis

assumes

that it

makes sense to speak of a

finite

ratio

between

two infinite

totalities,

but this is

certainly

not

an

obvious

assumption

and

should

be

defended

by

explaining

why

the

classical

strictures

against the

infinite should be

set

aside. Hobbes's

critique

of Wallis on these

points

is well-founded. Infinitesimal

mathematics is notoriously obscure, and Hobbes draws attention to the

conceptual

difficulties

raised by talk of

infinitely

small magnitudes

which

are greater than zero but less than any

positive

magnitude.

This

is

not to say

that

all of his criticisms of Wallis are to

the

(32) These aspects

of

Hobbes's critique

of

Wallis and the method

of indivisibles are

explored in [Giorello,

1990], [Mancosu

and Valiati, 1991, 65-70], and [Robinet, 1990].

My

treatment of them

is

correspondingly

brief.

(33)

See

[Six

Lessons,

5 ; EW, 7, 297-330] and the Postscript

to

Hobbes's

Censura Brevis

Doctrinae Wallisianae

De Motu [OL, 5,

84-88] for

succinct statements of

these

problems.



188

Douglas

M.

Jesseph

point,

or that

infinitesimal

methods

are irredeemable. But such

critiques are not the ravings of a madman.

Hobbes launches another attack

on Wallis

when he

notes

that

his

method

relies upon arithmetic "inductions" which are

transparently

invalid. As was observed above, Wallis

begins

his

quadratures with arithmetic results

concerning

infinite series. However,

he takes a general result as established

by simply

examining a small

number of

cases and insisting that

the general

case

holds "by

induction .

Thus,

in

Proposition

I of the

Arithmetica

Infinitorum he

declares:

The simplest

method

of investigation in

this and

in

some

following problems

is to consider

some

individual

cases

and to

observe the emerging ratios,

and

then to compare them

with one

another,

so

that

a

universal proposition

can

be shown

by

induction [AI,

1;

Wallis,

1693-1699,

1, 365]. Hobbes ridicules this

reasoning

in

his

Six

Lessons: "Egregious logicians and

geometricians, that think an induction, without

a

numeration of all the

particulars

sufficient, to

infer

a conclusion

universal, and

fit

to

be received for a geometrical demonstration " [Six Lessons, 5;

EW, 7,

308.]

These "inductions"

are unquestionably

the weakest

link in Wallis's reasoning, and Hobbes rightly

points out the

problem,

namely that

a conclusion about

the

infinite case

cannot

be rigorously demonstrated

simply

by

listing

some

initial

cases.

The

problem that

Wallis

faces is that

of

establishing the

convergence

conditions for

an infinite series,

and

the

difficulties

encountered in this area were a

key

problem in the early history of the

calculus that was not solved until the nineteenth century.

From this investigation

of Hobbes's

writings

on the

method

of indivisibles

it is

clear

that he

was quite familiar

with

the method

and saw

it

as a

key

to the solution of important geometric

problems, even

if

he regarded Wallis's

interpretation of

the method

as mistaken. Note, however, that

Hobbes's

enthusiasm

for

Cava-

lieri's

use

of the

method

has

more

to

do with Cavalieri's

reticence

on foundational

issues

than any strong affinity

between

their

positive view

on the nature of

geometry. Because

Cavalieri avoided

any direct discussion of

the

problems

surrounding

the composition

of the

continuum, Hobbes was free to interpret his

talk of

indivisibles in

a manner consonant

with his own

views on the nature

of lines

and surfaces.

Thus,

Cavalieri's deliberate vagueness

regarding the foundations of his

method left

Hobbes

the option

of

reading

him

as an

ally

rather than an

adversary.

Cavalieri

would



Hobbes on

the

Methods

of

Modern Mathematics

189

no

doubt

have

rejected

Hobbes's

conception

of points

and

lines

as extended,

and

he

is unlikely to have been sympathetic to

the

Hobbesian campaign for the elimination of analytic geometry.

Nevertheless, Hobbes did

manage to see Cavalieri

as

one of

his

few

mathematical allies.

V.

— Conclusions

The

picture

that

emerges

from

our

consideration of

Hobbes's

relationship

to

the methods

of

modern mathematics is

considerably more

complex

and interesting

than one

might

first

have

expected. Although

Hobbes was not a great mathematician,

he

was

not

unaware

of the

mathematical developments

of the

seventeenth

century.

As we

have

seen,

he

imagined

that

his

metaphysical principles had put him

in

a position to solve

famous

outstanding problems, and his rejection

of analytic

geometry

grows

out of

his

concern

with

placing mathematics

on

a

metaphysically

secure foundation.

It

happened that Hobbes was mistaken about

matters

of considerable

mathematical importance. In particular,

his campaign against algebraic methods was

nothing

short of a

disaster for his own

mathematical

ambitions.

But

as mistaken as

Hobbes was

about the

relevance

of algebra

to mathematics, we

can

at

least understand

how he was

led

to think that algebra

could

add

nothing

to geometry. Furthermore, his work is not simply

a record of mistakes

and confusions:

the critique of

Wallis's use

of indivisibles, for example, is largely on the mark and highlights

some

of

the conceptual difficulties in infinitesimal mathematics.

Hobbes's

own conception

of

mathematical demonstration as

a

science grounded in the consideration

of

true

causes may strike

us as

eccentric,

not to

say

bizarre.

Ours is an age that does not

take the concept

of

mathematical causality very

seriously,

but this

does not mean that

Hobbes's

conception of scientific and

mathematical

knowledge is

completely incomprehensible.

In fact,

Hobbes's

account of

mathematics

has a certain coherence when we see

it

as part of his program for a restructuring of science and

philosophy.

If

all

genuine

knowledge

of

things

derives

from

knowledge

of their causes, then

mathematics

must

also

employ

causal

princi-



190 Douglas M.

Jesseph

ples.

In the

end, we can see

that

Hobbes

worked

hard to make

a place for

mathematics

within his metaphysical system. This

involved re-writing much

of Euclid

to make geometry a science

of body, but

it also

entailed a

reading

of Cavalieri which gave

the

method

of indivisibles a

curiously

Hobbesian twist.

Unfortunately for

Hobbes,

his dreams

of mathematical glory were

pure

fantasies, albeit fantasies encouraged by a philosophical

methodology. We now know that some mathematical problems are simply

unsolvable

and

will

forever remain so.

But Hobbes's

continued

belief

in

his theory of geometry

as

the royal road

to

the solution

of all mathematical problems

led

him badly astray. In defending

himself against Wallis's criticisms, Hobbes was

led

to repudiate

nearly

all

of mathematics,

and

this

(more than

anything

else) shows

the hopeless state of his mathematical enterprise. Like all grand

systems, Hobbes's was ultimately a failure, and his failure is

nowhere more

evident than

in

his mathematical work. But failures

of

this magnitude, like

the

ruins

of an

ancient

city, are still worth

investigating.

North Carolina State

University,

Douglas M.

Jesseph.

Dept. of Philosophy and Religion,

Raleigh

(USA)

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