AN ESSAY ON The Uſefulneſs of MATHEMATICAL LEARNING, &c.
[1]I AM glad to hear from you, that the ſtudy of the Mathematicks is Promoted and Encouraged among the Youth of your Ʋniverſity. The great influence, which theſe Sciences have on Philoſophy and all uſeful Learning, as well as the Concerns of the Publick, may ſufficiently recommend them to your choice and conſideration: and the particular advantages, which You of that place enjoy, give Us juſt reaſon to expect from You a ſuitable improve⯑ment in them. I have here ſent you ſome ſhort reflections upon the Ʋſefulneſs of Mathematical Learning, which may ſerve as an argument to incite you to a cloſer and more vigorous purſuit of it.
[2] In all Ages and Countries, where Learning hath prevailed, the Mathe⯑matical Sciences have been looked upon as the moſt conſiderable branch of it. The very name [...] implies no leſs; by which they were called either for their excellency, or becauſe of all the Sciences they were firſt taught, or becauſe they were judg'd to comprehend [...]. And amongſt thoſe, that are commonly reckoned to be the ſeven Li⯑beral Arts, four are Mathematical, to wit, Arithmetick, Muſick, Geometry, and Aſtronomy.
But notwithſtanding their Excellency and Reputation, they have not been taught nor ſtudy'd ſo univerſally, as ſome of the reſt; which I take to have pro⯑ceeded from the following cauſes: The averſion of the greateſt part of Mankind to ſerious attention and cloſe arguing; Their not comprehending ſufficiently the neceſſity or great uſefulneſs of theſe in other parts of Learning; An Opinion that this ſtudy re⯑quires a particular Genius and turn of Head, which few are ſo happy as to be Born with; And the want of Publick Encouragement, and able Maſters. For theſe, and perhaps ſome other reaſons, this ſtudy hath been [3] generally neglected, and regarded only by ſome few perſons, whoſe happy Ge⯑nius and Curioſity have prompted them to it, or who have been forced upon it by its immediate ſubſerviency to ſome particular Art or Office.
Therefore I think I cannot do better ſervice to Learning, Youth, and the Na⯑tion in general, than by ſhewing, That the Mathematicks of all parts of humane know⯑ledge, for the improvement of the Mind, for their ſubſerviency to other Arts, and their uſe⯑fulneſs to the Common-wealth, deſerve moſt to be encouraged. I know a diſcourſe of this nature will be offenſive to ſome, who, while they are ignorant of Mathe⯑maticks, yet think themſelves Maſters of all valuable Learning: but their diſ⯑pleaſure muſt not deterr me from de⯑livering an uſeful truth.
The advantages, which accrue to the Mind by Mathematical ſtudies, conſiſt chiefly in theſe things: 1ſt. In accuſtom⯑ing it to attention. 2dly. In giving it a ha⯑bit of cloſe and demonſtrative reaſoning. 3dly. In freeing it from prejudice, credu⯑lity, and ſuperſtition.
Firſt, the Mathematicks make the Mind attentive to the objects, which it [4] conſiders. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not ob⯑vious. Truth is the ſame thing to the underſtanding, as Muſick to the ear, and Beauty to the eye. The purſuit of it does really as much gratifie a natural faculty implanted in us by our wiſe Creator, as the pleaſing of our Senſes: only in the former caſe, as the Object and Faculty are more Spiritual, the de⯑light is the more pure, free from the re⯑gret, turpitude, laſſitude, and intempe⯑rance, that commonly attend ſenſual pleaſures. The moſt part of other Sci⯑ences conſiſting only of probable rea⯑ſonings, the Mind has not where to fix; and wanting ſufficient principles to pur⯑ſue its ſearches upon, gives them over as impoſſible. Again, as in Mathematical inveſtigations truth may be found, ſo it is not always obvious: This ſpurs the Mind, and makes it diligent and atten⯑tive. In Geometria ſays Quinctilian, (lib. I. cap. 10.) partem fatentur eſſe utilem te⯑neris aetatibus: agitari namque animos, atque acui ingenia, & celeritatem percipiendi ve⯑nire inde concedunt. And Plato (in Repub. lib. VII.) obſerves, that the Youth, who [5] are furniſhed with Mathematical know⯑ledge, are prompt and quick at all other Sciences, [...]. Therefore he calls it [...]. And indeed Youth is generally ſo much more delighted with Mathematical ſtudies, than with the unpleaſant tasks, that are ſome times impoſed upon them, that I have known ſome reclaimed by them from idleneſs and neglect of learn⯑ing, and acquire in time a habit of thinking, diligence, and attention; qua⯑lities, which we ought to ſtudy by all means to beget in their deſultory and roving Minds.
The ſecond advantage, which the Mind reaps from Mathematical knowledge, is a habit of clear, demonſtrative, and metho⯑dical Reaſoning. We are contriv'd by Nature to learn by Imitation more than by Precept: And I believe in that re⯑ſpect Reaſoning is much like other in⯑feriour Arts (as Dancing, Singing, &c.) acquired by practice. By accuſtoming our ſelves to Reaſon cloſely about quan⯑tity, we acquire a habit of doing ſo in other things. It is ſurprizing to ſee, what ſuperficial, inconſequential Rea⯑ſonings, ſatisfie the moſt part of Man⯑kind. [6] A piece of wit, a jeſt, a ſimile, or a quotation of an Author, paſſes for a mighty Argument: with ſuch things as theſe are the moſt part of Authors ſtuffed: and from theſe weighty pre⯑miſes they infer their concluſions. This weakneſs and effeminacy of Mankind in being perſwaded where they are delight⯑ed, have made them the ſport of Ora⯑tors, Poets, and Men of wit. Thoſe lu⯑mina Orationis are indeed very good di⯑verſion for the Fancy, but are not the proper buſineſs of the Underſtanding; and where a Man pretends to write on abſtract ſubjects in a Scientifical method, he ought not to debauch in them. Lo⯑gical precepts are more uſeful, nay, they are abſolutely neceſſary for a rule of formal arguing in publick diſputations, and confounding an obſtinate and per⯑verſe adverſary, and expoſing him to the audience, or readers. But in the ſearch of truth, an imitation of the me⯑thod of the Geometers will carry a Man further than all the Dialectical rules. Their Analyſis is the proper model we ought to form our ſelves upon, and imi⯑tate in the regular diſpoſition and gra⯑dual progreſs of our enquiries; and even [7] he, who is ignorant of the nature of Ma⯑thematical Analyſis, uſes a method ſome⯑what Analogous to it. The Compoſition of the Geometers, or their method of de⯑monſtrating truths already found out, viz. by Definitions of words agreed upon, by Self-evident truths, and Propoſitions that have been already demonſtrated, is practicable in other ſubjects, tho' not to the ſame perfection, the natural want of evidence in the things themſelves not allowing it; but it is imitable to a conſiderable degree. I dare appeal to ſome writings of our own Age and Nation, the Au⯑thors of which have been Mathematically inclined. I ſhall add no more on this head, but that one, who is accuſtomed to the methodical Syſtems of truths, which the Geometers have reared up in the ſe⯑veral branches of thoſe Sciences, which they have cultivated, will hardly bear with the confuſion and diſorder of other Sciences, but endeavour as far as he can to reform them.
Thirdly, Mathematical knowledge adds a manly vigour to the Mind, frees it from prejudice, credulity, and ſuperſtition. This it does two ways, 1ſt. by accuſtom⯑ing us to examine, and not to take things [8] upon truſt. 2dly. By giving us a clear and extenſive knowledge of the Syſtem of the World; which, as it creates in us the moſt profound reverence of the Al⯑mighty and wiſe Creator; ſo it frees us from the mean and narrow thoughts, which ignorance and ſuperſtition are apt to beget. How great an enemy Mathe⯑maticks are to ſuperſtition, appears from this, That in thoſe Countries, where Ro⯑miſh Prieſts exerciſe their barbarous Ty⯑ranny over the minds of Men, Aſtrono⯑mers, who are fully perſwaded of the mo⯑tion of the Earth, dare not ſpeak out: But tho the Inquiſition may extort a Re⯑cantation, the Pope and a general Coun⯑cil too will not find themſelves able to perſwade to the contrary Opinion. Per⯑haps, this may have given occaſion to a calumnious ſuggeſtion, as if Mathematicks were an enemy to Religion, which is a ſcandal thrown both on the one and the other; for truth can never be an enemy to true Religion, which appears always to the beſt advantage, when it is moſt examined.
On the contrary, the Mathematicks are [9] friends to Religion; inaſmuch as they charm the paſſions, reſtrain the impetu⯑oſity of imagination, and purge the Mind from error and prejudice. Vice is error, confuſion and falſe Reaſoning; and all truth is more or leſs oppoſite to it. Be⯑ſides, Mathematical ſtudies may ſerve for a pleaſant entertainment for thoſe hours, which young Men are apt to throw away upon their Vices; the delightfulneſs of them being ſuch, as to make ſolitude not only eaſy, but deſirable.
What I have ſaid may ſerve to recom⯑mend Mathematicks for acquiring a vi⯑gorous Conſtitution of Mind; for which purpoſe they are as uſeful, as exerciſe is for procuring Health and Strength to the Body. I proceed now to ſhew their vaſt extent and Uſefulneſs in other parts of knowledge. And here it might ſuffice to tell you, that Mathematicks is the Sci⯑ence of quantity, or the Art of Reaſon⯑ing about things that are capable of more and leſs, and that the moſt part of the ob⯑jects of our knowledge are ſuch: as mat⯑ter, ſpace, number, time, motion, gra⯑vity, &c. We have but imperfect ideas of things without quantity, and as im⯑perfect a one of quantity it ſelf without [10] the help of Mathematicks. All the vi⯑ſible works of God Almighty are made in number, weight, and meaſure; therefore to conſider them, we ought to under⯑ſtand Arithmetick, Geometry, and Staticks: and the greater advances we make in thoſe Arts, the more capable we are of conſidering ſuch things, as are the ordi⯑nary objects of our Conceptions. But this will farther appear from particulars.
And firſt, if we conſider, to what per⯑fection we now know the Courſes, Pe⯑riods, Order, Diſtances, and Proportions of the ſeveral great Bodies of the Uni⯑verſe, at leaſt ſuch as fall within our view; we ſhall have cauſe to admire the Sagacity and Induſtry of the Mathema⯑ticians, and the power of Numbers and Geometry well apply'd. Let us caſt our Eyes backward, and conſider Aſtronomy in its Infancy: or rather let us ſuppoſe it ſtill to begin; for inſtance, a Colony of Rude Country people, tranſplanted into an Iſland remote from the commerce of all Mankind, without ſo much as the know⯑ledge of the Kalendar, and the Periods of the Seaſons, without Inſtruments to make Obſervations, or any the leaſt notion of Obſervations or Inſtruments. When is it, [11] we could expect any of their poſterity ſhould arrive at the Art of predicting an Eclipſe? Not only ſo, but the Art of reckoning all Eclipſes that are paſt or to come, for any number of Years? When is it, we could ſuppoſe, that one of thoſe Iſlanders tranſported to any other place of the Earth, ſhould be able by the in⯑ſpection of the Heavens to find how much he were South or North, Eaſt or Weſt of his own Iſland, and to conduct his Ship back thither? For my part, tho' I know this may be, and is daily done, by what is known in Aſtronomy; yet when I con⯑ſider the vaſt Induſtry, Sagacity, multi⯑tude of Obſervations, and other extrin⯑ſick things neceſſary for ſuch a ſublime piece of knowledge, I ſhould be apt to pronounce it impoſſible, and never to be hoped for. Now we are let ſo much in⯑to the knowledge of the Machine of the Univerſe, and motion of its parts by the Rules of this Science, perhaps the inven⯑tion may ſeem eaſy. But when we re⯑flect, what Penetration and Contrivance were neceſſary to lay the foundations of ſo great and extenſive an Art, we cannot but admire its firſt Inventors: as Thales Mileſius, who, as Diogenes Laertius and [12] Pliny ſay, firſt predicted Eclipſes; and his Scholar Anaximander Mileſius, who found out the Globous Figure of the Earth, the Aequinoctial Points, the Obli⯑quity of the Ecliptick, the principles of Gnomonicks, and made the firſt Sphere or Image of the Heavens; and Pythagoras, to whom we owe the diſcovery of the true Syſtem of the World, and order of the Planets. Tho' it may be, they were aſſiſted by the Egyptians and Chaldeans. But whoever they were, that firſt made theſe bold ſteps in this Noble Art, they deſerve the praiſe and admiration of all future Ages.
But tho' the induſtry of former Ages had diſcovered the Periods of the great [13] Bodies of the Univerſe, and the true Syſtem and Order of them, and their Orbits pretty near; yet was there one thing ſtill reſerved for the glory of this Age, and the honour of the Engliſh Na⯑tion, The grand ſecret of the whole Ma⯑chine; which, now it is diſcovered, proves to be (like the other contrivances of In⯑finite Wiſdom) ſimple and natural, de⯑pending upon the moſt known and moſt common property of matter, viz. gravity. From this the incomparable Mr. Newton has demonſtrated the Theories of all the Bodies of the Solar Syſtem, of all the primary Planets and their ſecondaries, and among others, the Moon, which ſeem'd moſt averſe to numbers: And not only of the Planets, the ſloweſt of which compleats its Period in leſs than half the Age of a Man, but likewiſe of the Co⯑mets, ſome of which its probable ſpend more than 2000. years in one Revolution about the Sun; for whoſe Theory he has laid ſuch a foundation, that after Ages aſſiſted with more Obſervations, may be able to Calculate their returns. In a word, the preceſſion of the Aequinoctial Points, the Tydes, the unequal Vibration of Pendulous Bodies in different Lati⯑tudes, [14] &c. are no more a queſtion to thoſe, that have Geometry enough to underſtand, what he has delivered on thoſe Subjects: A perfection in Philoſophy, that the boldeſt thinker durſt hardly have hoped for; and, unleſs Mankind turn barbarous, will continue the Reputation of this Nation, as long as the Fabrick of Nature ſhall en⯑dure. After this, what is it, we may not expect from Geometry join'd to Obſervati⯑ons and Experiments?
The next conſiderable object of Na⯑tural knowledge, I take to be Light. How unſucceſsful enquiries are about this Glorious Body without the help of Geometry, may appear from the empty and frivolous diſcourſes and diſputations of a ſort of Men, that call themſelves Phi⯑loſophers; whom nothing will ſerve for⯑ſooth, but the knowledge of the very Nature, and intimate Cauſes of every thing: while on the other hand, the Geometers not troubling themſelves with thoſe fruitleſs enquiries about the Na⯑ture of Light, have diſcovered two re⯑markable properties of it, in the refle⯑ction and refraction of its beams: and from thoſe, and their ſtreightneſs in other caſes, have invented the noble [15] Arts of Opticks, Catoptricks, and Dioptricks; teaching us to manage this ſubtile Body for the improvement of our knowledge, and uſeful purpoſes of Life. They have likewiſe demonſtrated the cauſes of ſe⯑veral Coeleſtial appearances, that ariſe from the inflection of its Beams, both in the Heavenly Bodies themſelves and other Phoenomena, as Parhelia, the Iris, &c. and by a late Experiment they have diſcovered the celerity of its motion. And we ſhall know yet more ſurprizing properties of Light, when Mr. Newton ſhall be pleas'd to gratifie the World with his Book of Light and Colours.
The Fluids which involve our Earth, viz. Air and Water, are the next great and conſpicuous Bodies, that Nature pre⯑ſents to our view: And I think we know little of either, but what is owing to Mechanicks and Geometry. The two chiefeſt properties of Air, its Gravity and Elaſtick force, have been diſcovered by Mechanical Experiments. From thence the decreaſe of the Air's denſity according to the increaſe of the diſtance of the Earth has been demonſtrated by Geometers, and confirmed by Experiments of the ſubſidence of the Mercury in the [16] Torricellian Experiment. From this like⯑wiſe, by aſſiſtance of Geometry, they have determined the height of the Atmo⯑ſphere, as far as it has any ſenſible den⯑ſity; which agrees exactly with another Obſervation of the duration of the Twi⯑light. Air and Water make up the object of the Hydroſtaticks, tho' denominated only from the latter, of which the prin⯑ciples were long ſince ſettled and demon⯑ſtrated by Archimedes, in his Book [...], where are demonſtrated the cauſes of ſeveral ſurprizing Phoenomena of Nature, depending only on the Aequi⯑librium of Fluids, the relative Gravities of theſe Fluids, and of Solids ſwimming or ſinking therein. Here alſo the Ma⯑thematicians conſider the different Preſ⯑ſures, Reſiſtances, and Celerities of So⯑lids moved in Fluids: from whence they explain a great many appearances of Nature, unintelligible to thoſe who are ignorant of Geometry.
Next, if we deſcend to the Animal Kingdom, there we may ſee the brighteſt ſtrokes of Divine Mechanicks. And whi⯑ther we conſider firſt the Animal Oeconomy in general, either in the internal motion and circulation of the Juices forced [17] through the ſeveral Canals by the motion of the Heart, or their external motions, and the Inſtruments wherewith theſe are performed, we muſt reduce them to Me⯑chanical Rules, and confeſs the ne⯑ceſſity of the knowledge of Mechanicks to underſtand them, or explain them to others. Borelli in his excellent Treatiſe de motu Animalium, Steno in his admirable Myologiae ſpecimen, and other Mathema⯑tical Men on the one hand, and the non⯑ſenſical, unintelligible ſtuff that the com⯑mon Writers on theſe Subjects have filled their Books with on the other, are ſuf⯑ficient inſtances to ſhew, how neceſſary Geometry is in ſuch ſpeculations. The only Organ of an Animal Body, whoſe ſtructure and manner of operation is ful⯑ly underſtood, has been the only one, which the Geometers have taken to their ſhare to conſider. It's incredible, how ſillily the greateſt and ableſt Phyſicians talked of the parts of the Eye and their uſe, and of the modus viſionis, before Kepler by his Geometry found it out, and put it paſt diſpute, tho' they apply'd themſelves particularly to this, and va⯑lued themſelves on it: and Galen pre⯑tended a particular Divine Commiſſion [18] to treat of it. Nay, notwithſtanding the full diſcovery of it, ſome go on in co⯑pying their Predeceſſors, and talk as Ʋn⯑geometrically as ever. It's true, we can⯑not reaſon ſo clearly of the internal mo⯑tions of an Animal Body, as of the ex⯑ternal, wanting ſufficient data and de⯑ciſive Experiments: But what relates to the latter (as the Articulation, Structure, Inſertion, and Vires of the Muſcles) is as ſubject to ſtrict Mathematical diſquiſi⯑tion, as any thing whatſoever; and even in the Theory of Diſeaſes and their Cures, thoſe, who talk Mechanically, talk moſt intelligibly. Which may be the reaſon for the Opinion of the ancient Phyſicians, that Mathematicks are neceſ⯑ſary for the ſtudy of Medicine it ſelf, for which I could bring long quotations out of their works. Among the Letters that are aſcrib'd to Hippocrates, there is one to his Son Theſſalus, recommending to him the ſtudy of Arithmetick and Geome⯑try, as neceſſary to Medicine. Galen in his Book intituled [...], begins, [...] [19] [...]. If one of the reaſons of the Ancients for this be now ſomewhat unfaſhionable, to wit, becauſe they thought a Phyſician ſhould be able to know the ſituation and aſpects of the Stars, which they believed had in⯑fluence upon Men and their Diſeaſes, (and poſitively to deny it, and ſay, that they have none at all, is the effect of want of Obſervation) we have a much better and undoubted one in its room; viz. That Mathematicks are found to be the beſt Inſtrument of promoting natural know⯑ledge. 2dly. If we conſider, not only the Animal Oeconomy in general, but likewiſe the wonderful ſtructure of the different ſorts of Animals, according to the different purpoſes for which they were deſign'd, the various Elements they inhabit, the ſeveral ways of procuring their nouriſhment, and propagating their kind, the different enemies they have, and accidents they are ſubject to, here is [20] ſtill a greater need of Geometry. It is pity, that the qualities of an expert Ana⯑tomiſt and skillful Geometer have ſeldom met in the ſame perſon. When ſuch a one ſhall appear, there is a whole Terra incognita of delightful knowledge to em⯑ploy his time, and reward his induſtry.
As for the other two Kingdoms; Bo⯑relli and other Mathematical Men, ſeem to have talked very clearly of Vegetation: and Steno another Mathematician, in his excel⯑lent Treatiſe de Solido intra Solidum natura⯑liter contento, has apply'd this part of learn⯑ing very handſomely to Foſſils and ſome other parts of Natural Hiſtory. I ſhall add only one thing more, That if we conſider motion it ſelf, the great Inſtru⯑ment of the Actions of Bodies upon one another, the Theory of it is entirely owing to the Geometers; who have de⯑monſtrated its Laws both in hard and elaſtick Bodies; ſhew'd how to meaſure it's quantity, how to compound and re⯑ſolve the ſeveral forces, by which Bodies are agitated, and to determine the Lines, which thoſe compound forces make them deſcribe: of ſuch forces gravity, being the moſt conſtant and uniform, affords a great variety of uſeful know⯑ledge, [21] in conſidering ſeveral motions that happen upon the Earth; viz. As to the free deſcent of heavy Bodies; The curve of projectiles; The deſcent and weight of heavy Bodies when they lye on inclined plains; The Theory of the mo⯑tion of Pendulous Bodies, &c.
From what I have ſaid, I ſhall draw but one Corollary, That a natural Phi⯑loſopher without Mathematicks is a very odd ſort of a perſon, that reaſons about things that have Bulk, Figure, Motion, Number, Weight, &c. without Arithme⯑tick, Geometry, Mechanicks, Staticks, &c. I muſt needs ſay, I have the laſt con⯑tempt for thoſe Gentlemen, that pretend to explain how the Earth was framed, and yet can hardly meaſure an Acre of Ground upon the ſurface of it: And as the Philoſopher ſpeaks, Qui repente pedi⯑bus illotis ad Philoſophos divertunt, non hoc eſt ſatis, quod ſint omninò [...] ſed legem etiam dant, quâ Phi⯑loſophari diſcant.
The uſefulneſs of Mathematicks in ſe⯑veral other Arts and Sciences is fully as plain. They were looked upon by the ancient Philoſophers as the key to all knowledge. Therefore Plato wrote upon [22] his School, [...], Let none unskilled in Geometry enter; and Xeno⯑crates told one ignorant in Mathematicks, who deſired to be his Scholar, that he was fitter to Card Wooll, [...], you want the handle of Phi⯑loſophy, viz. Geometry. There is no un⯑derſtanding the works of the ancient Philoſophers without it. Theo Smyrnoeus has wrote a Book entituled, An explana⯑tion of thoſe things in Mathematicks, that are neceſſary for the reading of Pla⯑to: Ariſtotle illuſtrates his precepts and other thoughts by Mathematical exam⯑ples, and that not only in Logick, &c. but even in Ethicks, where he makes uſe of Geometrical and Arithmetical pro⯑portion, to explain commutative and diſ⯑tributive juſtice.
Every body knows, that Chronology and Geography are indiſpenſable preparations for Hiſtory: a relation of matter of fact being a very lifeleſs inſipid thing with⯑out the circumſtances of time and place. Nor is it ſufficient for one, that would underſtand things thoroughly, that he knows the Topography, that is, the name of the Country, where ſuch a place lies, with thoſe of the near adjacent [23] places, and how theſe lie in reſpect of one another; but it will become him likewiſe to underſtand the Scientifical principles of the Art: that is, to have a true Idea of a place, we ought to know the relation it has to any other place, as to the diſtance and bearing, its Climate, Heat, Cold, length of days, &c. which things do much enliven the Readers no⯑tion of the very action it ſelf. Juſt ſo, it is neceſſary to know the Technical or Doctrinal part of Chronology, if a Man would be throughly skill'd in Hiſtory, it being impoſſible without it, to unravel the confuſion of Hiſtorians. I remem⯑ber Mr. Hally has determin'd the day and hour of Julius Coeſar's Landing in Britain, from the circumſtances of his relation. And every body knows, how great uſe our incomparable Hiſtorian Mr. Dodwell has made of the Calculated times of E⯑clipſes, for ſettling the times of great Events, which before were as to this eſſential circumſtance almoſt fabulous. Both Chronology and Geography, and alſo the knowledge of the Sun's and Moon's motions, ſo far as they relate to the conſtitution of the Kalendar and Year, are neceſſary to a Divine, and how ſadly [24] ſome otherwiſe Eminent have blunder'd, when they meddled with things that re⯑late to theſe, and border on them, is too apparent.
No body, I think, will queſtion the in⯑tereſt, that Mathematicks have in Paint⯑ing, Muſick, and Architecture, which are all founded on numbers. Perſpective and the Rules of Light and Shadows are owing to Geometry and Opticks: And I think thoſe two comprehend pretty near the whole Art of Painting, except deco⯑rum and ordinance; which are only a due obſervance of the Hiſtory and Circum⯑ſtances of the ſubject, you repreſent. For by Perſpective, may be underſtood the Art of deſigning the outlines of your ſolid, whether that be a Building, Land⯑skip, or Animal: and the draught of a Man is really as much the Perſpective of a Man, as the draught of a Building is of a Building; tho' for particular reaſons, as becauſe it conſiſts of more crooked lines, &c. it is hard to reduce the Perſpe⯑ctive of the former, to the ordinary eſta⯑bliſhed Rules.
If Mathematicks had not reduced Mu⯑ſick to a regular Syſtem, by contriving its Scales, it had been no Art, but Enthuſi⯑aſtick [25] Rapture, left to the roving fancy of every Practitioner. This appears by the extraordinary pains, which the Anci⯑ents have taken to fit numbers to three ſorts of Muſick, the Diatonick, Chroma⯑tick, and Enharmonick: which if we con⯑ſider with their nicety in diſtinguiſhing their ſeveral Modes, we ſhall be apt to judge, they had ſomething very fine in their Muſick, at leaſt for moving the paſ⯑ſions with ſingle Inſtruments and Voices. But Muſick had been imperfect ſtill, had not Arithmetick ſtepped in once more, and Guido Aretinus by inventing the temperament, making the Fifth falſe by a certain determined quantity, taught us to Tune our Organs, and intermix all the three kinds of the Ancients; to which we owe all the Regular and Noble Har⯑mony of our modern Muſick.
As for Civil Architecture (of Military I ſhall ſpeak afterwards) there is hardly any part of Mathematicks, but is ſome way ſubſervient to it. Geometry and Arithmetick for the due meaſure of the ſeveral parts of a Building, the Plans, Models, computation of Materials, time and charges: for ordering right its Arches and Vaults, that they may be [26] both firm and beautiful: Mechanicks for its ſtrength and firmneſs, tranſport⯑ing and raiſing materials: and Opticks for the Symmetry and Beauty. And I would not have any aſſume the character of an Architect without a competent skill in all of theſe. You ſee that Vi⯑truvius requires theſe and many more for making a compleat Architect. I muſt own, that ſhould any one ſet up to practice in any of the fore-mentioned Arts, furniſhed only with his Mathema⯑tical Rules, he would produce but very clumſy pieces. He, that ſhould pretend to draw by the Geometrical Rules of Per⯑ſpective, or Compoſe Muſick meerly by his skill in Harmonical numbers, would ſhew but aukward performances. In thoſe Compos'd Subjects, beſides the ſtiff Rules, there muſt be Fancy, Genius, and Habit. Yet nevertheleſs theſe Arts owe their being to Mathematicks, as laying the foundation of their Theory, and afford⯑ing them Precepts, which being once invented, are ſecurely rely'd upon by Practitioners. Thus many deſign, that know not a [...]ittle of the reaſon of the Rules, they practice b [...] and many no better quality [...] in their way Compoſe [27] Muſick, better perhaps than he could have done, that invented the Scale, and the Numbers upon which their Harmony is founded. As Mathematicks laid the foundation of theſe Arts, ſo they muſt improve them: and he, that would invent, muſt be skill'd in Numbers. Beſides it is fit a Man ſhould know the true grounds and reaſons of what he ſtudies: and he that does ſo, will certainly practice in his Art with greater judgement and variety, where the ordinary Rules fail him.
I proceed now to ſhew the more im⯑mediate uſefulneſs of Mathematicks in Civil Affairs. To begin with Arithmetick, it were an endleſs task to relate its ſeve⯑ral uſes in publick and private buſineſs. The regulation and quick diſpatch of both, ſeem intirely owing to it. The Nations, that want it, are altogether bar⯑barous, as ſome Americans, who can hardly reckon above twenty. And I be⯑lieve it would go near to ruine the Trade of the Nation, were the eaſy practice of Arithmetick aboliſhed: for example, were the Merchants and Tradeſmen oblig'd to make uſe of no other than the Roman way of notation by Letters, inſtead of our preſent. And if we ſhould feel the [28] want of our Arithmetick in the eaſieſt Calculations, how much more in thoſe, that are ſome thing harder; as Intereſt ſimple and compound, Annuities, &c. in which, it is incredible, how much the or⯑dinary Rules and Tables influence the diſpatch of buſineſs. Arithmetick is not only the great Inſtrument of private Commerce, but by it are (or ought to be) kept the publick Accounts of a Na⯑tion: I mean thoſe, that regard the whole State of a Common-wealth, as to the number, fructification of its people, in⯑creaſe of Stock, improvement of Lands and Manufactures, Ballance of Trade, Publick Revenues, Coynage, Military power by Sea and Land, &c. Thoſe, that would judge or reaſon truely about the State of any Nation, muſt go that way to work, ſubjecting all the fore-menti⯑oned particulars to Calculation. This is the true Political knowledge. In this reſpect the affairs of a Common-wealth differ from thoſe of a private Family, only in the greatneſs and multitude of particulars, that make up the accounts. Machiavel goes this way to work in his account of different Eſtates. What Sir William Petty and ſeveral others of our [29] Country-men have wrote in Political Arithmetick, does abundantly ſhew the pleaſure and uſefulneſs of ſuch Specula⯑tions. It is true, for want of good infor⯑mation, their Calculations ſome times proceed upon erroneous ſuppoſitions: but that is not the fault of the Art. But what is it, the Government could not perform in this way, who have the com⯑mand of all publick Records?
Laſtly, Numbers are applicable even to ſuch things, as ſeem to be govern'd by no rule, I mean ſuch as depend on Chance: The quantity of probability and propor⯑tion of it in any two propoſed caſes being ſubject to Calculation as much as any thing elſe. Upon this depend the princi⯑ples of Game. We find Sharpers know enough of this, to cheat ſome men that would take it very ill to be thought Bub⯑bles: And one Gameſter exceeds another, as he has a greater ſagacity and readineſs in Calculating his probability to win or loſe in any propoſed caſe. To under⯑ſtand the Theory of Chance throughly, requires a great knowledge of Numbers, and a pretty competent one of Algebra.
The ſeveral uſes of Geometry are not much fewer than thoſe of Arithmetick. [30] It is neceſſary for aſcertaining of pro⯑perty both in Plains and Solids, or in Surveying and Guaging. By it Land is ſold by the meaſure as well as Cloth: Work-men are pay'd the due price of their labour, according to the ſuperficial or ſolid meaſure of their work: and the quantity of liquors determined for a due regulation of their price and duty. All which do wonderfully conduce to the eaſy diſpatch of buſineſs, and the pre⯑venting of frauds and controverſies. I need not mention the Meaſuring di⯑ſtances, laying down of Plans and Maps of Countries, in which we have daily Experience of its uſefulneſs. Theſe are ſome familiar inſtances of things, to which Geometry is ordinarily apply'd: of its uſe in Civil, Military, and Naval Ar⯑chitecture we ſhall ſpeak afterwards.
From Aſtronomy we have the regular diſpoſition of our time, in a due ſucceſ⯑ſion of years, which are kept within their limits as to the return of the Sea⯑ſons, and the motion of the Sun. This is no ſmall advantage for the due repe⯑tition of the ſame work, Labour and Actions. For many of our Publick, Pri⯑vate, Military, and Country Affairs, Ap⯑pointments, [31] &c. depending on the pro⯑ducts of the Ground, and they on the Seaſons; It is neceſſary, that the returns of them be adjuſted pretty near to the motion of the Sun: and we ſhould quickly find the inconveniency of a vague unde⯑termined year, if we uſed that of the Mahumetans, whoſe beginning and every month wanders through all the days of ours or the Solar year, which ſhews the Seaſons. Beſide, the adjuſting of the Moon's motion to the Sun's is required for the decent Obſervation and Celebra⯑tion of the Church-Feaſts and Faſts accord⯑ing to the Ancient Cuſtom and Primitive Inſtitution; and likewiſe for the know⯑ing of the Ebbing and Flowing of the Tides, the Spring and Neap Tides, Cur⯑rents, &c. So that what-ever ſome peo⯑ple may think of an Almanack where all theſe are ſet down, it is oftentimes the moſt uſeful paper that is publiſhed the ſame year with it: Nay, the Nation could better ſpare all the Voluminous Authors in the Term-Catalogue, than that ſingle ſheet. Beſides, without a regular Chronology, there can be no certain Hiſtory; which appears by the confuſion amongſt Hiſtorians before the right diſ⯑poſition [32] of the year, and at preſent a⯑mong the Turks, who have the ſame confuſion in their Hiſtory as in their Ka⯑lendar. Therefore a matter of ſuch im⯑portance might well deſerve the care of the Great Emperour, to whom we owe our preſent Kalendar; who was himſelf a great proficient in Aſtronomy. Pliny has quoted ſeveral things from his Books of the Riſing and Setting of the Stars, Lib. XVIII. cap. 25, 26, &c. and Lucan makes him ſay,
The Mechanicks have produced ſo many uſeful Engines, ſubſervient to conveni⯑ency, that it would be a task too great to relate the ſeveral ſorts of them: ſome of them keep Life it ſelf from being a bur⯑den. If we conſider ſuch, as are invented for raiſing weights, and are employ'd in Building and other great works, in which no impediment is too great for them; or Hydraulick Engines for raiſing of Wa⯑ter, ſerving for great uſe and comfort to Mankind, where they have no other way to be ſupply'd readily with that ne⯑ceſſary Element; or ſuch as, by making [33] Wind and Water work for us, ſave Ani⯑mal force and great charges, and per⯑form thoſe actions, which require a vaſt multitude of hands, and without which every Man's time would be too little to prepare his own Aliment and other ne⯑ceſſaries; or thoſe Machines, that have been invented by Mankind for delight and curioſity, imitating the motions of Animals, or other works of Nature; we ſhall have reaſon to admire and extoll ſo excellent an Art. What ſhall we ſay of the ſeveral Inſtruments, which are con⯑triv'd to meaſure time? We ſhould quick⯑ly find the value of them, if we were re⯑duced to the condition of thoſe barba⯑rous Nations, that want them. The Pendulum-Clock invented and compleated by that famous Mathematician Monſieur Hugens is an uſeful invention. Is there any thing more wonderful than ſeveral Planetary Machines, which have been in⯑vented to ſhew the motions of the Hea⯑venly Bodies, and their places at any time? Of which the moſt Ingenious, ac⯑cording to the exacteſt Numbers and true Syſtem, was made by the ſame M. Hugens: to which we may very juſtly apply Claudian's noble Verſes upon that of Archimedes.
Here I ought to mention the Sciathe⯑rical Inſtruments, for want of which there was a time, when the Grecians themſelves were forced to meaſure the Shadow, in order to know the Hour; and as Pliny (cap. ult. lib. VII.) tells us, the Romans made uſe of an erroneous Sun-dial for ninety nine years, till Q. Marcius Philip⯑pus their Cenſor ſet up a better; which no doubt at that time was thought a Jewel. And at laſt, that famous Pyramid was ſet up in the Campus Martius, to ſerve for a Gnomon to a Dial marked on the [35] ſtreet. To this ſort of Engines ought to be referred Spheres, Globes, Aſtrolabes, Pro⯑jections of the Sphere, &c. Theſe are ſuch uſeful and neceſſary things, that alone may recommend the Art, by which they are made. For by theſe we are able in our Cloſet to judge of the Celeſtial mo⯑tions, and to viſit the moſt diſtant places of the Earth, without the fatigue and danger of Voyages; to determine con⯑cerning their diſtance, Situation, Climate, Nature of the Seaſons, length of their days, and their relation to the Celeſtial Bodies, as much as if we were Inhabi⯑tants. To all theſe I might add thoſe Inſtruments, which the Mathematicians have invented to execute their own pre⯑cepts, for making Obſervations either at Sea or Land, Surveying, Gauging, &c.
The Catoptricks and Dioptricks furniſh us with variety of uſeful inventions, both for the promoting of knowledge, and the conveniencies of Life; whereby Sight, the great Inſtrument of our per⯑ception, is ſo much improv'd, that nei⯑ther the diſtance, nor the minuteneſs of the Object are any more impediments to it. The Teleſcope is of ſo vaſt uſe, that, beſides the delightful and uſeful purpoſes [36] it is apply'd to here below, as the deſ⯑crying Ships, and Men, and Armies at a diſtance, we have by its means diſco⯑vered new parts of the Creation, freſh inſtances of the ſurprizing Wiſdom of the Adorable Creator. We have by it diſcovered the Satellites of Jupiter, the Satellites and Ring of Saturn, the Rota⯑tion of the Planets about their own Axes; beſides other appearances, where⯑by the Syſtem of the World is made plain to ſenſe, as it was before to reaſon. The Teleſcope has alſo improv'd the man⯑ner of Aſtronomical Obſervations, and made them much more accurate, than it was poſſible for them to be before. And theſe improvements in Aſtronomy, have brought along with them (as ever) cor⯑reſpondent improvements in Geography. From the Obſervation of Jupiter's Satel⯑lites, we have a ready way to determine the Longitude of places on the Earth. On the other hand, the Microſcope has not been leſs uſeful in helping us to the ſight of ſuch Objects, as by their mi⯑nuteneſs eſcape our naked eye. By it Men have purſued Nature into its moſt retir'd receſſes; ſo that now it can hard⯑ly any more hide its greateſt Myſteries [37] from us. How much have we learned by the help of the Microſcope of the contrivance and ſtructure of Animal and Vegetable Bodies, and the compoſition of Fluids and Solids? But if theſe Sci⯑ences had never gone further, than by their ſingle Specula and Lentes to give thoſe ſurprizing appearances of Objects and their Images, and to produce heat unimitable by our hotteſt Furnaces, and to furniſh infallible, eaſy, cheap, and ſafe remedies for the decay of our Sight ariſing commonly from old Age, and for purblindneſs; they had merited the great⯑eſt eſteem, and invited to the cloſeſt ſtudy: eſpecially if we conſider, that ſuch as na⯑turally are almoſt blind, and either know not their neareſt acquaintance at the diſtance of a rooms breadth, or cannot read in order to paſs their time pleaſant⯑ly, are by Glaſſes adapted to the defect of their Eyes ſet on a level again with thoſe that enjoy their Eye-ſight beſt, and that without danger, pain, or charge.
Again, Mathematicks are highly ſervice⯑able to a Nation in Military Affairs. I believe this will be readily acknowledg'd by every body. The Affairs of War take in Number, Space, Force, Diſtance, [38] Time, &c. (things of Mathematical con⯑ſideration) in all its parts, in Tacticks, Caſtrametation, Fortifying, Attacquing, and Defending. The Ancients had more oc⯑caſion for Mechanicks in the Art of War than we have: Gun-powder readily pro⯑ducing a force far exceeding all the En⯑gines, they had contriv'd for Battery. And this I reckon has loſt us a good occaſion of improving our Mechanicks: the cunning of Mankind never exerting it ſelf ſo much, as in their Arts of de⯑ſtroying one another. But, as Gun-powder has made Mechanicks leſs ſerviceable to War; it has made Geometry more neceſ⯑ſary: There being a force or reſiſtance in the due meaſures and proportions of the Lines and Angles of a Fortification, which contribute much towards its ſtrength. This Art of Fortification has been much ſtudy'd of late, but I dare not affirm, that it has attain'd its utmoſt perfection. And tho', where the ground is regular, it admits but of ſmall variety, the meaſures being pretty well deter⯑mined by Geometry and Experience, yet where the ground is made up of natu⯑ral Strengths and Weakneſſes, it affords ſome ſcope for thinking and contrivance. [39] But there is another much harder piece of Geometry, which Gun-powder has given us occaſion to improve, and that is the doctrine of Projectiles; whereon the Art of Gunnery is founded. Here the Geometers have invented a beautiful Theory, and Rules and Inſtruments, which have reduced the caſting of Bombs to great exactneſs. As for Tacticks and Caſtrametation, Mathematicks retain the ſame place in them as ever. And ſome tolerable skill in theſe are neceſſary for Officers, as well as for Engineers. An Of⯑ficer, that underſtands Fortification, will caeteris paribus much better defend his poſt, as knowing, wherein its ſtrength con⯑ſiſts, or make uſe of his advantage to his Enemy's Ruine, than he that does not. He knows, when he leads never ſo ſmall a party, what his advantages and diſad⯑vantages in Defending and Attacquing are, how to make the beſt of his Ground &c. And hereby can do truely more ſervice than another of as much Cou⯑rage, who, for want of ſuch knowledge, it may be, throws away himſelf and a number of brave Fellows under his Command: and it's well, if the miſchief reaches no further. As for a competent [40] skill in Numbers, it is ſo neceſſary to Of⯑ficers, that no Man can be ſafely truſted with a Company, that has it not. All the buſineſs is not to fire Muſquets; the managing of Affairs, the dealing with Agents, &c. happen more frequently. And the higher the Command is, the more skill in all the aforeſaid things is required. And I dare appeal to all the Nations in Europe, whether caeteris pari⯑bus Officers are not advanced in pro⯑portion to their skill in Mathematical Learning; except, that ſome times Great Names and Quality carry it; but ſtill ſo, as that the Prince depends upon a Man of Mathematical Learning, that is put as director to the Quality, when that Learn⯑ing is wanting in it.
Laſtly, Navigation which is made up of Aſtronomy and Geometry, is ſo noble an Art, and to which Mankind owes ſo many advantages, that upon this ſingle account thoſe Excellent Sciences deſerve moſt of all to be ſtudy'd, and merit the greateſt encouragement from a Nation, that owes to it both its Riches and Se⯑curity. And not only does the Com⯑mon Art of Navigation depend on Ma⯑thematicks, but whatever improvements [41] ſhall be made in the Architectura Navalis or Building of Ships, whether they are deſign'd for Merchant-Ships, or Ships of War, whether ſwift running, or bearing a great ſail, or lying near the wind be deſired, theſe muſt all be the improve⯑ments of Geometry. Ship-Carpenters in⯑deed are very induſtrious; but in theſe things they acknowledge their inability, confeſs that their beſt productions are the effects of chance, and implore the Geometers help. Nor will common Geo⯑metry do the buſineſs; it requires the moſt abſtruſe to determine the different ſections of a Ship, according as it is de⯑ſign'd for any of the foreſaid ends. A French Mathematician P. Le Hoſte has lately endeavoured ſome thing in this way: and tho' it is not free from errors, as requiring a fuller knowledge in Geo⯑metry; yet is the Author much to be commended for this, as having bravely deſign'd, and pav'd the way for other Mathematicians; and alſo for the former and bigger part of his Book, wherein he brings to a ſyſtem, the working of Ships, and the Naval Tacticks, or the re⯑gular diſpoſition of a Fleet in Attac⯑quing, Fighting and Retreating, accord⯑ing [42] to the different circumſtances of Wind, Tides, &c.
The great objection, that is made a⯑gainſt the neceſſity of Mathematicks, in the fore-mention'd great Affairs of Na⯑vigation, the Art Military, &c. is, that we ſee thoſe Affairs are carry'd on and managed by ſuch, as are not great Ma⯑thematicians; as Sea-men, Engineers, Surveyers, Gaugers, Clock-makers, Glaſs⯑grinders, &c. and that the Mathemati⯑cians are commonly Speculative, Retir'd, Studious Men, that are not for an active Life and buſineſs, but content themſelves to ſit in their Studies, and pore over a Scheme or a Calculation. To which there is this plain and eaſy anſwer: The Ma⯑thematicians have not only invented and order'd all the Arts above-mentioned, by which thoſe grand Affairs are ma⯑naged; but have laid down Precepts, contriv'd Inſtruments and Abridgements ſo plainly, that common Artificers are capable of practiſing by them, tho' they underſtand not a tittle of the grounds, on which the Precepts are built. And in this they have conſulted the good and neceſſities of Mankind. Thoſe Af⯑fairs demand ſo great a number of peo⯑ple [43] to manage them, that it is impoſſible to breed ſo many good or even tole⯑rable Mathematicians. The only thing then to be done was to make their Pre⯑cepts ſo plain, that they might be under⯑ſtood and practiſed by a multitude of Men. This will beſt appear by exam⯑ples. Nothing is more ordinary than diſpatch of buſineſs by common Arith⯑metick, by the Tables of ſimple and com⯑pound Intereſt, Annuities &c. Yet how few Men of buſineſs underſtand the reaſons of common Arithmetick, or the contri⯑vance of thoſe Tables, now they are made; but ſecurely rely on them as true. They were the good and the Thorough-Mathematicians, that made thoſe Precepts ſo plain, and Calculated thoſe Tables, that facilitate the practice ſo much. Nothing is more univerſally neceſſary, than the meaſuring of Plains and Solids: And it is impoſſible to breed ſo many good Mathematicians, as that there may be one, that underſtands all the Geometry requiſite for Surveying, and meaſuring of Priſms and Pyramids, and their parts, and meaſuring Fruſtums of Conoids and Spheroids, in every Market-Town, where ſuch work is neceſſary: [44] the Mathematicians have therefore in⯑ſcrib'd ſuch Lines on their common Rulers, and Slipping Rulers, and adapted ſo plain Precepts to them, that every Country-Carpenter, and Gauger, can do the buſineſs accurately enough; tho' he knows no more of thoſe Inſtruments, Tables, and Precepts he makes uſe of, than a Hobby-horſe. So in Navigation, it is impoſſible to breed ſo many good Mathematicians, as would be neceſſary to ſail the hundredth part of the Ships of the Nation. But the Mathematicians have laid down ſo plain and diſtinct Precepts, Calculated neceſſary Tables, and contriv'd convenient Inſtruments, ſo that a Sea-man, that knows not the truths, on which his Precepts and Tables depend, may practice ſafely by them. They reſolve Triangules every day, that know not the reaſon of any one of their Operations. Sea-men in their Calculations make uſe of artificial Numbers or Loga⯑rithmes, that know nothing of their con⯑trivance: and indeed all thoſe great inventions of the moſt famous Mathema⯑ticians had been almoſt uſeleſs for thoſe common and great Affairs, had not the practice of them been made eaſy to thoſe [45] who cannot underſtand them. From hence it is plain, that it is to thoſe Spe⯑culative Retir'd Men, we owe the Rules, the Inſtruments, the Precepts for uſing them, and the Tables which facilitate the diſpatch of ſo many great Affairs, and ſupply Mankind with ſo many conveni⯑encies of Life. They were the Men, that taught the World to apply Arithmetick, Aſtronomy, and Geometry to Sailing, with⯑out which the needle would be ſtill uſe⯑leſs. Juſt the ſame way in the other parts of Mathematicks, the Precepts that are practiſed by multitudes, without be⯑ing underſtood, were contriv'd by ſome few great Mathematicians.
Since then it has been ſhewn, how much Mathematicks improve the Mind, how ſubſervient they are to other Arts, and how immediately uſeful to the Common-wealth, there needs no other arguments or motives to a Government, to encou⯑rage them. This is the natural conclu⯑ſion from theſe premiſes. Plato in his Republick (lib. VII.) takes care, That, who⯑ever is to be Educated for Magiſtracy, or any conſiderable Poſt in the Common-wealth, may be inſtructed firſt in Arithmetick, then in Geometry, and thirdly in Aſtronomy. [46] And however neceſſary thoſe Arts were in Plato's time, they are much more ſo now: The Arts of War and Trade re⯑quiring much more the aſſiſtance of thoſe Sciences now, than they did then; as be⯑ing brought to a greater height and per⯑fection. And accordingly we ſee, theſe Sciences are the particular care of Princes, that deſign to raiſe the Force and Power of their Countries. It is well known, that this is none of the leaſt Arts, where⯑by the French King has brought his ſub⯑jects to make that Figure at Sea, which they at this time do; I mean, the care He takes for Educating thoſe appointed for Sea-ſervice in Mathematical Learning. For in the Ordonnance Marine Title VIII. ‘'He orders, that there be Profeſſors to teach Navigation publickly in all the Sea-port Towns, who muſt know de⯑ſigning, and teach it to their Scholars, in order to lay down the appearances of Coaſts, &c. They are to keep their Schools open, and read four times a week to the Sea-men, where they muſt have Charts, Globes, Spheres, Com⯑paſſes, Quadrants, Aſtrolabes, and all Books and Inſtruments neceſſary to teach their Art. The directors of Hoſ⯑pitals [47] are oblig'd to ſend thither yearly two or three of their boys to be taught, and to furniſh them with Books and Inſtruments. Thoſe Profeſſors are oblig'd to examine the Journals depo⯑ſited in the Office of Admiralty, in the place of their eſtabliſhment; to correct the errours in preſence of the Sea-men, and to reſtore them within a month, &c.'’ King Charles the ſecond, who well un⯑derſtood the importance of Eſtabliſh⯑ments of this nature, founded one ſuch School in Chriſt's Hoſpital London; which, I believe, is inferiour to none of the French: but 'tis to be wiſhed there were many more ſuch. His preſent Majeſty, during the time of the late War, Eſta⯑bliſhed a Mathematical Lecture to breed up Engineers and Officers, as knowing very well the importance thereof. And this continued ſome time after the Peace. And it is worthy the conſideration of the Wiſdom of the Nation, whether the reſtor⯑ing and continuing this, even in Peace, be not expedient for the breeding of En⯑gineers, who are ſo uſeful and valuable, and ſo difficult to be had in time of War, and ſo little dangerous in times of Peace.
[48] Beſides the crowd of Merchants, Sea-men, Surveyors, Engineers, Ship-carpenters, Artiſans, &c. that are to be inſtructed in the practice of ſuch parts of Mathe⯑maticks, as are neceſſary to their own buſineſs reſpectively, a competent num⯑ber of able Mathematicians ought to be entertained, in order to apply themſelves to the practice; not only to inſtruct the former ſort, but likewiſe to remove thoſe obſtacles, which ſuch, as do not think beyond their common Rules, can⯑not overcome. And no doubt it is no ſmall impediment to the advancement of Arts, that Speculative Men and good Ma⯑thematicians are unacquainted with their particular defects, and the ſeveral cir⯑cumſtances in them, that render things practicable or impracticable. But if there were publick encouragement, we ſhould have skilful Mathematicians employed in thoſe Arts, who would certainly find out and remedy the imperfections of them. The preſent Lords Commiſſio⯑ners of the Admiralty knowing, that there are ſtill two great Deſiderata in Na⯑vigation, to wit, The Theory of the varia⯑tion of the magnetical Needle, and a method of finding out the Longitude of any place, [49] that may be practicable at Sea by Sea-men, and being ſenſible, of what im⯑portance it would be to find out either of them, have imployed a very fit per⯑ſon, the ingenious Mr. Hally, who has joyn'd an entire acquaintance in the practice, to a full and thorough know⯑ledge of the more abſtruſe parts of Ma⯑thematicks. And now that he is return⯑ed, it is not doubted, but he will ſatisfy thoſe, that ſent him, and in due time the World too with his diſcoveries in both thoſe particulars, and in many other, that he has had occaſion to make. And where a long ſeries of Obſervations and Experiments is neceſſary, he has no doubt laid ſuch a foundation, as that After-Obſervers may gradually perfect them. If it were not for more than the correcting the ſituation of the Coaſts, where he touched, and by them others, whoſe relation to the former is known, the Nation is more then triply pay'd; and thoſe, who ſent him, have by this Miſſion ſecured to themſelves more true Honour and laſting Fame, than by Actions, that at firſt view appear more Magnificent.
[50] The next thing, that is neceſſary for the improvement of Mathematical Learn⯑ing, is, That Mathematicks be more ge⯑nerally ſtudy'd at our Ʋniverſities than hitherto they have been. From thoſe Seminaries the State juſtly expects and demands thoſe, who are acquainted both with the Speculation and Practice. In thoſe are all the encouragements to them imaginable, Leiſure and Aſſiſtance. There are ſtill at hand Books and Inſtruments, as alſo other Scholars that have made equal progreſs, and may be Comrades in ſtudy, and the direction of the Pro⯑feſſors. There are alſo in perfection all the incitements to this ſtudy, and eſpeci⯑ally an acquaintance with the works of the Ancients, where this Learning is ſo much recommended: There other Facul⯑ties are ſtudy'd, to which it is ſubſervient. There alſo are the Nobility and Gentry bred, who, in due time muſt be called to their ſhare in the Government of the Fleets, Army, Treaſury, and other Publick Employments, where Mathematical Learn⯑ing is abſolutely neceſſary, and without which, they, tho' of never ſo great Natural parts, muſt be at the mercy and diſcretion of their Servants and Depu⯑ties; [51] who will firſt cheat them, and then laugh at them. And not only Publick Employments, but their Private Con⯑cerns demand Mathematical knowledge. If their Fortunes lie in Woods, Coal, Salt, Manufactures, &c. the neceſſity of this knowledge is open and known: and even in Land-Eſtates, no undertaking for improvement can be ſecurely rely'd upon without it. It not only makes a Man of Quality and Eſtate his whole Life more Illuſtrious, and more uſeful for all Affairs, (as Hippocrates ſays, [...] ) but in particular, it is the beſt Companion for a Country Life. Were this once become a faſhionable ſtudy (and the Mode exerciſes its Em⯑pire over Learning as well as other things) it is hard to tell, how far it might influence the Morals of our No⯑bility and Gentry, in rendring them Se⯑rious, Diligent, Curious, taking them off from the more fruitleſs and airy exer⯑ciſes of the Fancy, which they are apt to run into.
[52] The only Objection, I can think of, that is brought againſt theſe ſtudies, is, that Mathematicks require a parti⯑cular turn of Head, and a happy Genius that few people are Maſters of, without which all the pains beſtowed upon the ſtudy of them are in vain: They ima⯑gine that a Man muſt be Born a Mathema⯑tician. I anſwer, that this Exception is common to Mathematicks and other Arts. That there are perſons, that have a par⯑ticular capacity and fitneſs to one more than another, every body owns: And from experience I dare ſay, it is not in any higher degree true concerning Ma⯑thematicks than the others. A Man of good ſenſe and application is the per⯑ſon, that is by nature fitted for them: eſpecially if he begins betimes; And if his circumſtances have been ſuch, that this did not happen, by prudent directi⯑on the defect may be ſupply'd as much as in any Art whatſoever. The only advantage this Objection has, is, that it is on the ſide of ſoftneſs and idleneſs, thoſe powerful Allies.
There is nothing further remains, Sir, but that I give you my thoughts in ge⯑neral concerning the Order and Method [53] of ſtudying Mathematicks; which I ſhall do very ſhortly, as knowing that you are already acquainted with the beſt me⯑thods, and others with you may have them eaſily from the beſt and ableſt hands.
Firſt then, I lay down for a princi⯑ple, that no body at an Ʋniverſity is to be taught the practice of any rule without the true and ſolid reaſon and demonſtration of the ſame. Rules with⯑out demonſtration muſt and ought to be taught to Sea-men, Artiſans, &c. as I have already ſaid; and Schools for ſuch people are fit in Sea-ports and Trading-Towns; but it is far below the dignity of an Ʋniverſity, which is deſign'd for ſolid and true Learning, to do this. It is from the Univerſities, that they muſt come, who are able to remedy the de⯑fects of the Arts: and therefore no⯑thing muſt be taken on truſt there. Sea-men and Surveyors, &c. remember their Rules, becauſe they are perpetu⯑ally practiſing them: But Scholars, who are not thus employ'd, if they know not the demonſtration of them, preſently forget them.
[54] Secondly, no part of Mathematicks ought to be taught by Compendiums. This follows from the former. Compen⯑diums are fit to give a general and ſu⯑perficial knowledge, not a thorough one. It's time, and not the bulk of Books, we ought to be ſparing of: And I appeal to any perſon of Experience, whether ſo⯑lid knowledge is not acquir'd in ſhorter time by Books treating fully of their ſubjects, than by Compendiums and A⯑bridgements.
From hence it follows, that the Ele⯑ments of Arithmetick and Geometry are to be taught. Euclid in his thirteen Books of Elements gives us both: but our preſent way of Notation ſuperſedes ſome of thoſe of Arithmetick, as demon⯑ſtrating the Rules from the Operations themſelves. There remain then the firſt ſix Books for the Geometry of Plains, and the laſt three for Stereometry. The reſt ought to be read in their own place for the perfection of Arithmetick. In teaching theſe, care ought to be taken to make uſe of ſuch Examples, as ſuit with the condition of the Scholar. For inſtance, Merchants Accounts and Affairs [55] for Examples of the Operations of A⯑rithmetick, to one that is afterwards to have a concern that way; whereas to a Man of the firſt Quality, examples from the encreaſe and decreaſe of the peo⯑ple, or from Land or Sea-Force, and from the Tacticks ought to be propoſed. For it is certain, nothing makes one tyr'd ſooner, than the frivolous and trifling examples, that are commonly brought for the exerciſe of the Rules of Arith⯑metick and Geometry: tho' this is com⯑mon to them with the other Arts, as Grammar, Logick, &c.
The manner of Writing of the Ma⯑thematicians of this and the former Age makes Trigonometry, with the man⯑ner of conſtructing its Tables, &c. al⯑moſt Elementary; and the practical Geo⯑metry commonly ſo call'd, is very fit to come next, as an elegant application of the Elements of Geometry to Buſineſs, as Surveying, Gauging, &c.
After the Elements of Sphericks, which are perfectly well handled by Theodoſius, a full inſight into the principles of Aſtro⯑nomy will be neceſſary.
[56] Mechanicks come next to be read, which are the Ground of a great part of Natural Learning: and afterwards Op⯑ticks, Catoptricks and Dioptricks.
But none of theſe except the Elements can be fully underſtood until one is pret⯑ty well skill'd in Conick-ſections: And all theſe are made more eaſy by ſome to⯑lerable skill in Algebra, and its applica⯑tion to Geometry.
Theſe foundations being laid, any one may with great eaſe purſue the ſtudy of the Mathematicks, as his occa⯑ſions require: either in its abſtract parts, and the more recondite Geometry, and its application to Natural knowledge; or in Mechanicks, by proſecuting the Sta⯑ticks, Hydroſtaticks, Balliſticks, or in Aſtronomy, by its application to Geogra⯑phy, Navigation, Gnomonicks, Aſtrolabes, &c. But in moſt of theſe a particular order is not neceſſary. Any one may take that firſt, which he is moſt incli⯑ned to.
I ſhall not offer you any advice con⯑cerning the choice of Books, but refer you (if you want any) to the direction of thoſe, who are Eminent among you [57] in this part of Learning. I ask your pardon for the omiſſion of Ceremony in theſe papers, having followed rather the ordinary way of Eſſay than Letter: and wiſhing you good ſucceſs in your ſtudies, I am,