history of mathematics

History of Mathematics

Paul Yiu

Department of MathematicsFlorida Atlantic University

Spring 2014

1A: Pythagoras’ Theorem in Euclid’s Elements

Euclid’s Elements

An ancient Greek mathematical classic compiled in the Third century B. C.TheElementsconsists of 13 books.

I – VI Plane geometryVII – IX Number theoryX Theory of irrational constructible quantitiesXI–XIII Solid geometry

Yiu: History of Mathematics 2014

1

Euclid, Book I

Book I II III IV V VI TotalDefinitions 23 2 10 7 18 5 65Common notions 5Postulates 5Propositions 48 14 37 16 25 33 173


2

Euclid, Book I: The definitions

(1) A point is that which has no part.(2) A line is breadthless length.(4) A straight line is a line which lies evenly with the points on itself.(10) When a straight line standing on a straight line makesthe adjacent angles equal to one another, each of the equal angles isright ,and the straight line standing on the other is calledaperpendicular to that on which it stands.(11) An obtuse angle is an angle greater than a right angle.(12) An acute angle is an angle less than a right angle.(15) A circle is a plane figure contained by one linesuch that all the straight lines falling upon itfrom one point among those lying within the figureequal one another.(16) And the point is called thecenterof the circle.(23)Parallel straight lines are straight lines which,being in the same planeand being produced indefinitely in both directions,do not meet one another in either direction.


3

Euclid Book I: The postulates

(1) To draw a straight line from any point to any point.(2) To produce a finite straight line continuously in a straight line.(3) To describe a circle with any center and radius.(4) That all right angles equal one another.


4

Euclid Book I: The fifth postulates

(5) That, if a straight line falling on two straight linesmakes the interior angles on the same side less than two right angles,the two straight lines, if produced indefinitely,meet on that side on which are the angles less than the two right angles.


5

The common notions

(1) Things which equal the same thing also equal one another.(2) If equals are added to equals, then the wholes are equal.(3) If equals are subtracted from equals, then the remainders are equal.(4) Things which coincide with one another equal one another.(5) The whole is greater than the part.


6

Euclid I.1

On a given finite straight line to construct an equilateral triangle.

A B

C

D E

Construct the circlesC(A,B)andC(B,A) to intersect at a pointC. ThenABC is an equilateral tri-angle.


7

Euclid I.4

If two triangles have two sides equal to two sides respectively,and have the angles contained by the equal straight lines equal,then they also have the base equal to the base,the triangle equals the triangle,and the remaining angles equal the remaining angles respectively,namely those opposite the equal sides.

Congruence test SSS: △ABC ≡ △XY Z if

AB = XY, BC = Y Z, CA = ZX.

A

B C

X

Y

Z


8

Isosceles triangles I.5

An isoscelestriangle [is] that which has two of its sides alone equal,and ascalenetriangle [is] that which has its three sides unequal.1

Euclid I.5. In isosceles triangles the angles at the base equal one another,and, if the equal straight lines are produced further, then the angles underthe base equal one another.

A

B C

F G

D E

Proof. ExtendAB andAC toD andE re-spectively. Choose an arbitrary pointF onBD, and (by I.3) constructG onCE suchthatCG = BF .The trianglesAFC andAGB are congru-ent by I.4. It follows thatFC = GB and∠BFC = ∠AFC = ∠AGB = ∠CGB.Again, by I.4, the trianglesBFC andCGB are congruent.From this,∠CBF = ∠BCG,and∠ABC = ∠ACB.


1Euclid seems to take isosceles and scalene in the exclusive sense. But it is more convenient to take thesein the inclusive sense. An isosceles triangle is one with twoequal sides, so that an equilateral triangle is alsoisosceles.

9

Euclid I.6: converse of I.5

(6) If in a triangle two angles equal one another,then the sides opposite the equal angles also equal one another.


10

Euclid I.4,8,26 (congruence tests)

Euclid did not use the termcongruenceof triangles. When he says twotriangles are equal, he means they are equal in area.

Euclid I.4, SAS: If two triangles have two sides equal to two sides re-spectively, and have the angles contained by the equal straight lines equal,they will also have the base equal to the base, the triangle will be equal to thetriangle, and the remaining angles equal the remaining angles respectively,namely those which the equal sides subtend.

Euclid I.8, SSS: If two triangles have the two sides equal to two sidesrespectively, and have also the base equal to the base, they will have theangles equal which are contained by the equal straight lines.

Euclid I.26, ASA or AAS: If two triangles have two angles equal totwo angles respectively, and one side equal to one side, namely, either theside adjoining the equal angles, or that subtending one of the equal angles,they will also have the remaining sides equal to the remaining sides and theremaining angle to the remaining angle.

The RHS test is not in theElements.


11

Congruence tests

(1) SSS: △ABC ≡ △XY Z if

AB = XY, BC = Y Z, CA = ZX.

A

B C

X

Y

Z

(2) SAS: △ABC ≡ △XY Z if

AB = XY, ∠ABC = ∠XY Z, BC = Y Z.

A

B C

X

Y

Z


12

Congruence tests

(3) ASA: △ABC ≡ △XY Z if

∠BAC = ∠Y XZ, AB = XY, ∠ABC = ∠XY Z.

A

B C

X

Y

Z

(4) AAS. We have noted that this is the same asASA: △ABC ≡ △XY Z

if

∠BAC = ∠Y XZ, ∠ABC = ∠XY Z, BC = Y Z, .

A

B C

X

Y

Z


13

Congruence tests

(5) RHS. The ASS is not a valid test of congruence. Here is an example.The two trianglesABC andXY Z are not congruent even though

∠BAC = ∠Y XZ, AB = XY, BC = Y Z.

A X

B Y

C Z

A X

B Y

C Z

However, if the equal angles are rightangles, then the third pair of sides areequal:

AC2 = BC2−AB2 = Y Z2−XY 2 = XZ2,

andAC = XZ. The two triangles are congruent by the SSS test. With-out repeating these details, we shall simply refer to this as theRHS test.△ABC ≡ △XY Z if

∠BAC = ∠Y XZ = 90◦, BC = Y Z, AB = XY.


14

Parallel lines

Euclid I.27. If a straight line falling on two straight linesmakes the alternate angles equal to one another,then the straight lines are parallel to one another.

Euclid I.28. If a straight line falling on two straight linesmakes the exterior angle equal tothe interior and opposite angle on the same side,or the sum of the interior angles on the same side equal to two right angles,then the straight lines are parallel to one another.


15

Parallel lines

Euclid I.29. A straight line falling on parallel straight lines makes the alter-nate angles equal to one another, the exterior angle equal to the interior andopposite angle, and the sum of the interior angles on the same side equal totwo right angles.

Euclid I.30. Straight lines parallel to the same straight line are also parallelto one another.


16

Parallelograms

Euclid I.34. In parallelogrammic areas the opposite sides and angles areequal to one another, and the diameter bisects the areas.

BA

DC

Euclid I.35 Parallelograms which are on the same base and in the sameparallels are equal to one another.

B C

A D E F


17

Some constructions with parallel lines

Euclid I.31. Through a given point to draw a straight line parallel to a givenstraight line.

I.42. To construct, in a given rectlineal angle, a parallelogram equal to agiven triangle.

I.44. To a given straight line to apply, in a given rectilineal angle, a parallel-ogram equal to a given triangle.

I.45. To construct, in a given rectlineal angle, a parallelogram equal to agiven rectlineal figure.


18

Euclid I.36-41

I.36. Parallelograms which are on equal bases and in the same parallels areequal to one another.

I.37. Triangles which are on the same base and in the same parallels areequal to one another.

I.38. Triangles which are on equal bases and in the same parallels are equalto one another.

I.39. Equal triangles which are on the same base and on the same side arealso in the same parallels.

I.40. Equal triangles which are on equal bases and on the same side are alsoin the same parallels.

I.41. If a parallelogram have the same base with a triangle and be in thesame parallels, then the parallelogram is double of the triangle.


19

Euclid I.46: Construction of a square

Euclid I.46. To describe a square on a given straight line.

(1)

(9)

(8)C

A B

D

(2)(3)

(4) (5)

(6)

(7)


20

Euclid VI.3: Angle bisector theorem

A

B CD

E

If an angle of a triangle is bisected and thestraight line cutting the angle cuts the basealso, the segments of the base will havethe same ratio of the remaining sides of thetriangle; and if the segments of the basehave the same ratio as the remaining sidesof the triangle, the straight line joined fromthe vertex to the point of the section willbisect the angle of the triangle.


21

External angle bisector theorem

This is also true for the bisector of the external angle:If E is a point on the extension of the sideBC,AE bisects an external angle ofBAC

if and only ifBE : EC = AB : AC.

A

B CD E

Figure 1:

Corollary. LetB andC be given points.The locus of a pointPwhose distances fromB andC are in a constant ratio (not equal to 1)is a circle.


22

Euclid I.47

In right angled triangles the square on the side subtending the right angle isequal to the squares on the sides containing the right angle.

A

BC

There are many proofs of this fundamental theorem in geometry.The proof given by Euclid is importantbecause it gives a way of converting a rectangle into a square.


23

Euclid I.42,43

Euclid I.42. To construct, in a given rectilineal angle, a parallelogram equalto a given triangle.

D

A F G

B E C

Euclid I.43. In any parallelogram the complements of the parallelogramsabout the diameter are equal to one another.

K

B G C

A H D

E F


24

Euclid I.44,45

Euclid I.44. To a given straight line to apply,2 in a given rectilineal angle,a parallelogram equal to a given triangle.

D

C

A

BG

H

EF K

M

L

Euclid I.45. To construct, in a given rectilineal angle, a parallelogram equalto a given rectilineal figure.

B K H M

F G L

C

A

D

E

Figure 2: Euclid I.45


2The termapplication is explained in Proclus’ commentary: “application starts with one side given andconstructs the area along it, neither falling short of the length of the line nor exceeding it, but using it as oneof the sides enclosing the area”.

25

Proof of I.47

D E

K

H

G

F

A

BC

L

M

D E

K

H

G

F

A

BC

L

M

SquareABFG = 2 ·∆CBF

= 2 ·∆ABD

= rectangleBDLM.

Similarly, SquareACHK = rectangleCELM .Therefore, SquareABFG + squareACHK = squareBDEC.


26

Euclid I.48: converse of I.47

If in a triangle the square on one of the sides be equal to the squares on theremaining sides of the triangle, the angle contained by the remaining sidesof the triangle is right.

ac

b

a

bC X

B Y

A Z

Proof. Let ABC be a triangle for whichBC2 + CA2 = AB2. Considera right triangleXY Z with Y Z = BC, ZX = CA and∠XZY = a rightangle. By Euclid I.47,XY 2 = Y Z2 + ZX2 = BC2 + CA2 = AB2;XY = AB. Therefore, the trianglesABC andXY Z are congruent byEuclid I.8, and∠ACB = ∠XZY = a right angle.


27

A useful corollary of Euclid I.47

Let the perpendicular from the right angle vertexC of a right triangleABC

intersect the hypotenuseAB atX. Then

AC2 = AX · AB, BC2 = BX · BA.

A B

C

X


28

Euclid II.14

To construct a square equal to a given rectilineal figure.

BG FE

H

DC

Proof without words:

BG FE

H

DC

BG FE

H

DC


29

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