| IO T H E F R A N K L I N J O U R N A L AND

very readily, as we always have some to spare, they being prepared at ihe turn, ace with a wrought iron bar about 15 inches lonff~ towhich we weld the old handle, when we find it necessary to make use of a new brand. I n~ed make no further observations, than that the s~eps ~ff the .~pindles ougl~t to be cast of the hardest iron : and in re- lathm ~) .'!;e br:mds, that in casting, the end of the bar of wrought ir,m shouhl be ,~?/it in four parts fi~r an inch, and opened a little, ~nd the brand cast ~pon tli'at end, to prcvcnt their becoming loose, wllich they are likely to do without this precaution.

A ~ cast iron brand costs about ha~'as much, and will last about half the time, of one that is wrought, and cyst. But there is, in various respects, a great advantage in having two brands instead of one.

J. MOltTO~ P o o ~ .

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

By the late Mr. Jo~s Cltoss, Teacher of Y~]athematics, G&sgou'.

No. II .

In every a~e and crmntrv where learning has prevailed, tile l~,athe- matical scie~ices have been esteemed as fi~rming an extensixe and

• o t a . . . . wuunble, l~art of human, knowledge. . . . . . l he~r influence on the mind, m- de~)endent of their practical utility m hfe, ts ahme suftlcmnt to t~lace them on a level wi~h most other studios. They accustom the mind to alton inn. .In the i:ursuit of this study, we are delighted with a succe.~shm of com~ecwd truth% which al:e evident when we under- stand tl~e reasonings,, bu~ which do not appear at first sight to depend on each other. The i,'~suit t:ftruth, gratifies a faculty implanted in us. as much as the ptcas~i~g of our senses ; and the pleasure which we by ttmt means derive, is flee from the regret, the turpitude, and the intemperance which often attend sensual pleasures. When any one has it, It pleasure, he will naturally wish fin" a repetition of it ; but this he cannot i~ave, unless he understamls the reasoning by which a~y thing is shown to be true ; and as mathematical truths are not obviL ous, he witl be incited t~ study the reasoning by which any conclu- sion is aimed at. Attention is requisite fi)r this purpose; arid by en- deavouri~:g to attend closely to one subject, a habit of attention witl soon be established.

By mathematical knowledge we acquire a habit of clear, demo~l- strative, and me.thodieal reasnning. If we to~& into controversial writing, hear verbal disputes, or examine the fimndations of man)] in- genious systems, we shall be SUrl;rised at what superficial reasonings satisfy the greatest part of mankind. The method of convinciiig which is often adopted, is to work upon the passions, radmr than the judgment. A piece of wit, an anecdote, or a simile, is often advanced "in place of solid sense. You may hear a subject, abundantly plain, obscured by reasoning the most fonlishl and often you may see a man of plato common sense supporting a position~ which scarce re--

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~iuires proof~ silenced by tile noise or the laugh of his opponent, or knocked down by a quotation from Shakspear~' or Hudlbras' I f with this we compare the reasonin# of the ~eometriciau, we shah find it is conducted itia manner diamet~'ieallv t%e reverse. " ....

From a few ,simple axioms and self-evident .princiNes,. _ we .pr°ceed gradually to the most general propositions, anti remote analogies ; de- ducin;~; one truth from those already known, in a chain of reasoning intMliifly connected, antl logically pursued ; and truth a~reeably forces kself upon the mind, in the full title of irresistible conviction.

By an attachment to these studies, we acquire an elevation of thought which fixes the mind, and prepares it for other pursuits; we tbllow with pleasure the same closehess of argu nent-in other re- searches ; and if we find a want of this accuracy, our mind, accus- tumed to tie contemplation of truth undisguised, revolts indignant from the merely probable, the false or erroneous asserti.n, and reason disdains to assent to the sophisticated tale.

I t is true, indeed, that in most other sub.jetty, ihe sa,ne strictness of reasoning is imlwacticalfle , because their I)rinciph.s do not admit of the same do~ree of evldence ; but still i t ' s m~'a tble in a eer(am degree. For proof of t;lli~ I lleed only appeal to tl,,se works, the au- thors of which were mathematically inclined Of these 1 shall only

Smith s IA'ealth o! Nations, Rmd's Essays on the Powers of Man, and the writings of Dr. Beattie.

What has been said may recommend mathematics, as a useful ex- ercise to the mind ; the exercise of our mental powers is as necessary for giving a vigorous constitution of mind° as bodily exercise is tbr procuring health and strenath to our flame. But it is not merely in a speculative view that thg study of mathematics is advanta_~e;us; ~ts apphcatmn to other sciences and the arts, have gtven rise to many inventions, which are useful in every department of life.

Tim importance of mathematics as a preparation fi)r other sciences, will be e{qdent on considering their connexion and de[~endence on mathematical principles. We may observe, that all the objects of our knowledge a rd madh in number, weight, and measure, and therefore to consider them we ought to uuderstaml arithmetic, geometry, and ~tatics ; and the greater advances we make in those arts, we are the more capable of considering such things as are the ord!nary obj-cts of our conceptions. But particularly we now know, w~th asm,nshmg precision, the courses, periods, order, distances, and proportions of the several g 'eat bodies whic ~ comp(,sc our planetary s),~tem, at least of such as are within ore' view ; and this att~wds a rem~,rkabte instance of the power of arithmetic, a,td .~e;m~ctry~ u eli applied, l~et us sup- pose ,,)urselves |)laced at that sti,,ge of society, when the l(mg course of observ~,tlon and .~tudv IlecPssl)l'F to In'i,"g.c as(vo,,>my, ~<)}ts p:'e.~ent perfecti(,.n~ had yet lo be'~in ;--let us suppose omselve:, i~li(wa~)t ,)f the most obvious, revc!uti,ms alid motion,'; ef our t)!al:ci ;~Ic; u~ sUl,F,)se o,~a'sc, lve,s ig~:(,rant ',)f its periods and seasons, with,ut instrua)ents to l,l[tt,;O observations, without any hle:,_ ~f ()bs(.rvat~tms m " iIlstt'un)elll~;~ ---when would we exp.:.'ct tha~ a , y o~ oul po..te..t) shm, id arrive at {h{: art <,i ~.>rediel.ing a~ ec!i),~.e ? N, Vhc~, w,),ld ~c SUpl,,~e them ca-

I I ~ '~'H~ ~'aam~LIN a o m ~ a c A~rJ

pable of reckv~ino: all the eclipses, past, or' to crone, for any number of years ? When would we suppose that if conveyed to any dislance }k'om theh home, that they should be able to tell ]mw fat" from it they" were, sm~lh, north, east, or west, or what course to take to tettlFn ? We know that all this may be done, and is daily done, by what is known in astwm,>mv; ye~, When we consider the vast indusi, rv, saga- city, and the multitude .fobsevvations necessary fovthese purposes, we wo'ul(t be i~clhwd tu give up the puvsui~ as impossible tb be attained; but b,,' (:he aasistance of the mathe.matlcal selence% these things ave new rendered so easy, lhat they may be pet'fiu-umd by ordinary un- ders~amllngs.

What has been said of ash'onomy is equally true of geography and of navigation, sciences which depend upon astronomical and t~[athe- matical principles. 3'o chronology these sciencesare equally necessary; frm~ (he occu)'cence ef circumstances vektted in many parts of ancient hist(wy, the p)'ecise dates have often been ascertained when some re- matkable events have taken place. Thus M r. lIalley de(,cvmined the day and hour of Julius Cmsar's landing in Britain; and the accounts ot'~mcient eclipses, have, by ~his means, et~abled us tu tell exactly tim dates ~fi" alton', events in histo:v, wlfich, without this veriIication, might be reckoned Iiibulmls. ']'hus~ in order to read histvry with advantage, some knowledge of geoo/vaphv and chronohav are necessary.

Light is a consideratSle object of ttatural '~towledge ; but all inqui- ries concerning this hody, are frivolous and futile, unless guided by geometry. I am not to 'be u ldevstoud here as speak t~g of tte inge- nious theories of the chemists concerning this wonderful substance-- with these, lhe geometers have nothing to do; but they have discov- er."d tw~ of tim pt-ineipal laws of its action, viz. in the"reflectiou and refraction , f i t s beams: a~d have invented the be~.q~tiful theory ~f op- tics, atul ~fvellected and yet'ratted vision, and have laught us to ma- nage 'ii~is subtUe body, fin' the improvement of our knowledge, and tbr pm'pos'.~:~ useful in life. Geometers have likewise demonstrated the, causes of several eete:¢dal appearances which arise from the in- ltexi~;u of i{s beams~ both in the planets, &e. and in the phenomena which m'ise in the atmosphere of our earth. Of the fluids which sue- rout:d, or i}oat upola the surface of the globe which we inhabit, air and water, little could be ktiow~.t but 1"o," the assistance of geometry and mechanics. The e!astieity, and gravity. ~l" air have been discov- ered by mechanical exlJeriments. From these properties, geomeu'i- clans flare calculated the heigt~t of the atmvsphere, as far as it has any ,~ensible &rosily, and the result agrees with almther ob:;ervatin~t ~m tl:e du~'aiioa ~Jf the twiiight. Air amt water are the objects of hvdvo,~st;:tics, thouah denomilu~ted only from the latter. 'l't~e princi- ples of this scienct"", were esi:abtished'by Archimedes, and b)' the sei.. ence is illustrated thuse nat,wal appearances which depend on the gravity and nt,'>tion of fluids, and the motion of solids in these ttuids; in considering tl~e dit~i~reut presstlres~ celerities, and resistap.ces of these solids, many practical observatim~s t~ave been pointed out, ne- cessary tbr the business of naval architecture; and the solid, which :st~all i~ass through a fluid with the least possible resistance, has been

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ascertained. The calculations of mathematics on the motiQn ~wate r through pipes, &c. are useful for several purposes: I mi~h~ enume~ rate other arts and sciences whicla depend on rnathematicaYprinciples~ thus perspective depends upon the rules of~eometrv and ootics,i. The

athematlcs have reduced music to a regular systenb by inventing its scales ; and there is hardly any part of mathematics which is not sub~ servient io architecture. ~I might descend to the animal fraine' The eye, the only organ of sense w~ich the geometers have¢onsidered, i s the only one whose structure and manner of operatinn are understood.

Every anatomist who wnuld wish to understand the action of the bones and muscles, would need the knowledge of mechanics. .....

The usefulness of mathematics in several sciences and practical arts, is abundantly evident. They were reckoned by some of the an- cient philosophers the key to all knowledge. Thus 151ato had written above the door of his school. " I , e t no one ignorant of geometry enter here." I t is not meant, however, that a knowledge of mathematics will enable any person lo practise in any of the arts I have mention- ed. Thus he who would attempt to compose music merely by his knowledge of harmonical numbers, could not be expected t() [produce very excellent pieces. To excel in several of those arts, bestdes the accuracy of rules, a person must be possessed of genius and fancy, and practice is necessary fi)r them all. Yet still they owe their bein~ to mathematics, which" lays the tbundatiou of their theory, and afl~'rds them precepts, which being once invented, are depended upon secure- ly by practitioners. Thus, though many design who do not know the reas~ons of the rules by which they practise--though many compose music better, perhaps, than the inventor of the scale could t~ave done, and know nothing of the numbers on which their harmony is founded, yet as the mathematics Show the fou~latioa nf these a~rts~ they must be necessary for their improvement; and surely it Will be granted, that he who knows the fundamental principles of what he professe% has the best chance of excelling.

ttints to Paviers. t ly C0r~oxr.L MACXaO.~L [Continued from page 58.]

The next kinds of pavement that it may be necessary-to mention, are those of Florence, nf Sienna, of Milan, and some other cities of Northern Italy. These may, indeed, be assimilated to a kind of stone rail-road, as there are partlcu~a, tracku allotted for the wheels~ and others for the horses. [he track~ for tl~e wheels are composed of stones nf very large dimensions; they are of marble, lumacelar limestone, or of a very hard sand stone ; most of them, particularly at Florence, weighing several tons. They are laid wi~h much preci- sion, in lines of about three feet broad. The spades for the horses, between these lines, are paved with small stones, and are, as well as I can recollect, about four feet wide. In some of the squares, the small pavement predominate~; while the lines of large stone-way

VOL, I I Io~No. ~,~FEm~vAr~v, 18~7. 15