$0 ~ g raA~K~l~ Jo~aNa~ aN~

I,ondon durh~g the year 1824, !,505,021 chahh'ons of coal, to supply the consuniption of I, omlon anti its environs, which, at that tnn% were estimated to contain 1,500,000 inhabitants. This quantity gives very nearly one chaldron, or f ig bushels, of Newcastle coals ~average cost 48"shillings, or $10.67) for each person ; and the quan- tity of heat evolved from the combustion of which, is only equal to that from ~0 bushels of Lehigh coal. Now we find that in Philadd- phia about one ton, or 28 busliels of Lehigh coal, may be estimated to be required for each person; the proportion therefore between the two places for each inhabitant, in quantity, is as "20 to ~28, and in cost~ as 10.67 to 7.04. The discrepance in the quantity, may be at- tributed to the milder climate of London during the winter season, and to the greater frugality of its inhabitants, in the use of fuel, both in the arts, and in domestic economy.

Philadelphia, January 1, 18527.

The following notice is extracted from Professor 5~lllman's .Tournal, ¥ o l . xi. No . 1. p. 98.

I have been just favoured with a copy e f a Memoir, by Mr. Marcu~ ~ Bull, read betbre the American Philosophical Society of Philadelphia~ April "7, 1826--entitled :

" Ex.,:p.eriments to determine, the corn. ]~arative quantities, of. [[eat, evolved in the combustion of the prinezpal varieties of wood and coal, used in the United States for Fuel; and also to determine the compa- rative quantities .°f Heat lost by. the ordinar~y Tpa aratus made use of

for their combushon." This memoir is the result of a long course of experiments, evident-

lv conducted with great care and skill. It is replete with interesting information, and is to be regarded as one of the most i:nportal~t con- tril)utions o}- science to the arts and to domestic economy, which has been made for a hmg time in this country. I t is wor0)y of being carefldlv studied, both by scientific and practical men, and f-)r the sake of ' the latter class, it might be well if an analysis of this elabo- rate anti detailed paper, presenting, in a lucid aml concise form. the i,ractieal important results which have been obtained by Mr. Bull-- were prepared tbr extensive circulation,

"FROM T H E GLASGOW MV.CIIANIC~' MAGAZIXE,

ESSAYS ON M A T H E M A T I C S .

Ry the lale Mr. JoH:~ CRoss, Teacher of ~lthematics, Glasgow.

No. I.

The word mathematics originally signified discipline or learning, (science;) but it is now appropriated ~te tllat science which teaches

" - - - - ~ f r "

the comparison and mensuratmn of ma~,mtude. I t has ab;o been tanned tM sciunce of quantity, anti in this view, the objects to which

AMERIOAN ~IICHANI¢S ~ MAOAZ[NL 5 |

it ma~ be applied are equally numerous and various as the objects of our sense's'; for whate\,er ~'s the object of our senses, is capab-le of being considered, either with respect to its number, its extetision~ or its quantity.

Mathematics are divided into pure and mixed. Pure mathematics considers. . magnitudes generallv,~, simpl.y~ and

abstractedly, without any relahon to matter or sensible o b j e c t s . - Under this class are comprehended,

1~ Arithmetic, or the art of numerical computation; ~, Geometry~ which teaches us to measure extension ; 3, Analysis, or the comparison and calculation of magnitudes in

general ; 4, Mixed geometry, or the combination of geometry with analysis. Mixed mathematics borrow fi'om physics, that is natural philoso-

phy, one or more incontestable experimdnts, and then, by a demonstra- tive chain of reasoning, they deduce, from established principles, conclusions as certain as those of pm'e mathematics.~Under this division are comprehended,

1~ Mcchanlcs, the sclc~Jce which treats of the elYect of moving powers, or forces, and the laws of motion ;

~, tls'drodynamics, which explains the motion of fluids~ and the laws of~their action ;

3, Astronomy, which considers the revolutions and variou~ phe nomena of" the sun, moon, and other heavenly bodies ;

4, Optics, the science of vision~ including the properties of light and colours ;

5, Acoustics, the theory 0f sounds. M~tthematics have also been divided into speculative and practical ;

a division which appIies both to the pure and mixed. S l~eculative, mathematics. • in¢luires after knowledge, which it is .t'r°"

posed to attain, and snnply contemplates the truth or fMsehood of what is asserted.

.Practical mathematics is the application of the speculative, anti shows how to perfiwm something useful or advautageous to mankind.

i t is ~mt possible to iix the ocigin of mathematics with precision, though we are able to aflh'm that it goes back to the remotest a,_,es. Josephus asserts that they were studied belbre the th)od ; that~he sons of Seth were observers of the heavens; that thev built two pil- lars, the otae of brick, the other of stone, to commetr~orate their dis- coveries, and that Abraham taught these sciences to the Egyptians,

people known in history. I he tnagl, ov priests el Egypt, (hreeted by the laws of their insti

tution to study and collect the secrets of nature: were become ihe depositaries a.mt dispensers of all humat~ km)~.,led~e'; but they have been blamed ~xich involving their discoverie.~ ir~ my.,tery. IL i~ said

5~ TI~E raANX~.m JO UrnAL a.~D

that Thales travelled into Egypt about 600 years before Christ, and • brought the mathematical scmtmes into Greece ; and, whether this be true or not, it is only from this time that we have any certain ac- counts of them.

With the n~athematics of the ancients we are acquainted only through the writings of the Greeks, aml we do not possess the neces- sary documents to estimate the instruction which these deriwed from their intercourse with tile more eastern nations. We know, however, that as soon as the mathematics took root in Greece, their progress was rapid, and the Grecians became, in some measure, the preceptors of all nations. The elementary books collected and arranged by Euclid, }lave been t,'anslated into all languages, anti have continued. for more than ~000 ye'u-s to be exclusiveiy taught in eve,'y mathema- tical sehool~a certain proof of their excellence. The conics of Apollonius hold an equal rank in what has been called tile higher

~ eometry. Many other mathematicians laboured in the same field, ut tl{e most exalted rank in this legion of honour has been assigned,

both by reason and fable, to the sublime genius of" Archimedes. In the accurate ~cienee~ whidl require cool attention, silence, and

meditation, the Romans never surpassed mediocrity. Use- ss as the means of attaining the first offices of the state, they we, re

the occupation of a few obscure individuals, remote fi-om tl{e~, gitation of public affi6rs. The Roman mathematicians were little more than translators or commentators of Archimedes, Apollonius, &c.

On the death of Theodosius, and the dMsion of the empire, the western part was long ravaoed, and at length subjugated by barba- rians, and .soon sunk into °rofoun~[p. ignorance;. . ~'~hile the eastern ...~chools, were wholly em.i)loyed, m theolog, c a l . .dtsputes. The accurate sctences had taken refuge m the Atexandrian museum, (almost the olaly., refu,;e,, they had, left them,.) which was founded b ' ) PCo ett. y Phtladelphus, about S~0 years before Christ. Here the mathemattcs flourished neat- ten centuries. But of this asylum they were deprived about the middle of the seventh century, when the Arabs, conducted' by the successors of Mohammed, ~pread cat'nave and devastation through the east, the museum and library of Alexan~h'ia were destroyed.

However, though the chain of mathematical discovery was broken by this fatal eatastro,, phe,. a few links, remained,, which this nqtion, of destroyers, snitched by the charms of peace, strove to collect am[ unite afresh. In less than a century we find the Arabs cultivating astronomy, and their tas*e for a particular science gradually extend- ed to all the branches of knowledge. For the space of seven hun- dred years, the matlmmatics thmrished in the extensive dominions of the caliphs; by the Moors they were carried into Spain, and spread over the rest of Europe. And the Arabs rendered essential service by their translations of the works of the ancient Greeks, with some of whigh we are acquainted, only through the medium of the Arabic version.

The conquests of the Turks brough't back ignorance and barbarism, on the delightful countries wlfich the Arabs ini~abitedj and extin-

AMEa~CAN ~ECHANICS ~ M,~GAZI~E. ,55

~ u~shed the ]amp of science which had long glimmered with fading ~stre in the dissolute and desolated provinces of the eastern empire.

At the taking of Constantinople by Mohammed 1I. a persecution arose against artists and men of learning, by which many of them were destroyed; but some escaped~ and carried with then tile re- mains of the mathematical sciences into Italy, France, Germany, and England, e'ountries in which ~ taste for literature had already begun to take root.

From this period, all the branches of the mathematics made rapid progress. The improvements of the moderns have put us in posses- sion of an infinite number of problems, inaccessible to the ancient ~ eometrieians. The ancients }lave given us nearly the whole of what

as been termed pure or speculative geometry, few additions have been made to the fundameutals of the science~most of the modern works on plane geometry, or the conic sections, are compilations from theirs; but tile moderns have improved arithmetic. Tile ancients knew little, if any thing of algebra, and we claim the st/blime inven- tion ofttuxions ~{s our own. We have applied the mathematics to the improvenlent of practical arts and sciences, as mechanics, projec-

But let it not be supposed that the moderns have surpassed the an- cients in genius. It is probable that the discoveries which they made, required, in the infancy of science, *intellectual exertions as great as any modern improvement. Tile most important improve- ments in science have always been preceded by gradual advances. The discovery which astonishes when it iv announced, eau often be traced through many previous steps which all contribute to bring it to Iight; anti the wonder that it should not have been sooner made, is often greater than the surprise which it occasions.

~¥e are possessed of instruments, which are equally just, which arrive, by a shorter process, at the end proposed; but we are infe- rior to the ancients in pure geometry. If Archimedes were to return to lhis earth, with all the wonderful attainments he possessed, he would be obliged to subject himself to a long course of study before he could place himself on a level with Newton; yet Newton lamented that he had too much neglected tile strict geometrical reasonings ot the Syracusan sage.

On th~ coral reefs and islands of ,gustralasia ct~d Polynesia.

Throughout the whole range of the Polynesian and Australasian Islands, there is scarcely a league of sea mmccupied by a coral reef; or a coral island ; the ti~rmer springing up to the surf~tceof the water~ perpendicularly from the ikthomlcss bottom, "deeper than did ever plummet sound," and the latter in various stages, fi.om tile low and naked rock~ with the water rippling over it, to an uninterrupted forest