(baptised 29 Apr 1667 - 27 Feb 1735)
A GENTLEMAN in the CITY to his FRIEND at OXFORD.
The THIRD EDITION.
Booksellers in Ave-Mary-Lane, near St. Paul's, London. M.DCC.XLV.
I AM glad to hear from you, that the Study of the Mathematics is Promoted and Encouraged among the Youth of your University. The great Influence, which these Sciences have on the Philosophy, and all useful Learning, as well as the Concerns of the Public, 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 Improvement in them. I have here sent you some short Reflections upon the Usefulness of Mathematical Learning, which may serve as an argument to incite you to a closer and more vigorous Pursuit of it.
In all Ages and Countries, where Learning hath prevailed, the Mathematical Sciences have been looked upon as the most considerable Branch of it. The very Name Μάθησις 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 judged to comprehend πάντα τα Μαθήματε. [p.4] And, amongst those, that are commonly reckoned to be the Seven Liberal Arts, Four are Mathematical, to wit, Arithmetic, Music, Geometry, and Astronomy.
But, notwithstanding their Excellency and Reputation, they have not been taught nor study'd to universally, as some of the rest; which I take to have proceeded 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 requires a particular Genius and Turn of Head, which few are so happy as to be born with; And the Want of public Encouragement, and able Masters. For these, and perhaps some other Reasons, this Study hath been generally neglected, and regarded only by some Persons, whose happy Genius and Curiosity have prompted then to it, or who have been forced upon it by its immediate Subserviency to some particular Art of Office.
Therefore, I think I cannot do better Service to Learning, Youth, and the Nation in general, than by shewing, That the Mathematics, of all Parts of human Knowledge, for the Improvement of the Mind, for their Subserviency to other Arts, and their Usefulness to the Commonwealth, deserve most to be encouraged. I know a Discourse of this Nature will be offensive to some, who, while they are ignorant of Mathematics, yet think themselves Masters of all valuable Learning: But their Displeasure must not deter me from delivering an useful Truth.
The Advantages which accrue to the Mind by Mathematical Studies, consist chiefly in these Things: 1st, In accustoming it to Attention. 2dly, In giving it a Habit of close and demonstrative Reasoning. 3rdly, In freeing it from Prejudice, Credulity, and Superstition.
[p.5] First, the Mathematics make the Mind attentive to Objects, which it considers. This they do by entertaining it with a great Variety of Truths, which are delightful and evident, but not obvious. Truth is the same thing to the Understanding, as Music to the Ear, and Beauty to the Eye. The Pursuit of it does really as much gratify a natural Faculty implanted in us by our Creator, as the pleasing of our Senses: Only in the former Case, as the Object and Faculty are more spiritual, the Delight is the more pure, free from Regret, Turpitude, Lassitude, and Intemperance, that commonly attend sensual Pleasures. The most Part of other Sciences consisting only of probable Reasonings, the Mind has not where to fix; and, wanting sufficient Principles to pursue 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 attentive. In Geometria (says Quinctilian, lib. I. cap. 10) partem satentur esse utilem teneris aetatibus: agitari namque animos, atque acui ingenia, & celeritatem percipiendi venire inde concedunt. And Plato (in Repub. lib. VII.) observes, that the Youth, who are furnished with Mathematical Knowledge, are prompt and quick at all other Sciences, ειςωάυλα τα Μαθήαλα οξεις φαίυουλαι. Therefore, he calls it χαλα ωαιδείαυ οδόυ. And, indeed, Youth is generally so much more delighted with Mathematical Studies, than with the unpleasant Talks, that are sometimes imposed upon them, that I have known some reclaimed by them from Idleness, and Neglect of Learning; and acquire in time a Habit of Thinking, Diligence, and Attention; Qualities, 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 methodical Reasoning. We are [p.6] contrived by Nature to learn by Imitation more than Precept: And, I believe, in that respect, Reasoning is much like other inferior Arts (as Dancing, Singing, &c.) acquired by Practice. By accustoming ourselves to reason closely about Quantity, we acquire a Habit of doing so in other things. It is surprising to see, what superficial inconsequential Reasonings satisfy the most Part of Mankind. 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 Premises they infer their Conclusions. This Weakness and Effeminancy of Mankind in being persuaded where they are delighted, have made the Sport of Orators, Poets, and Men of Wit. Those lumina orationis are indeed very good Diversion for the Fancy, but are of 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 them. Logical Precepts are more useful, nay, are absolutely necessary, for a Rule of formal Arguing in public Dispositions, and confounding an obstinate and perverse Adversary, and exposing him to the Audience of Readers. But, in the Search of Truth, an imitation of the Method of Geometers will carry a Man farther than all Dialectical Rules. Their Analysis is the proper Model we ought to form ourselves upon, and imitate in the regular Disposition, and gradual Progress, of our Inquiries; and even he, who is ignorant of the Nature of Mathematical Analysis, uses a Method somewhat analogous to it. The Composition of the Geometers, or their Method of demonstrating 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 [p.7] Want of Evidence in the things themselves not allowing it; but it is imitable top a considerable Degree. I dare appeal to some Writings of our own Age and Nation, the Authors 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 several branches of those 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 Supersition. This it does in two Ways: 1st, By accustoming us to examine, and not to take things 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 almighty 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 Mathematics are to Superstition, appears from this, That in those Countries, where Romish Priests exercise their barbarous Tyranny over the Minds of Men, Astronomers, who are fully persuaded of the Motion of the Earth, dare not speak out: But tho' the Inquisition may extort a Recantation, the Pope, and a general Council too, will not find themselves able to persuade to the contrary Opinion. Perhaps, this may have given Occasion to calumnious Suggestion, as if Mathematics were an Enemy to true Religion, which appears always to the best Advantage, when it is most examined.
Te capiet magis. ------
[p.8] On the contrary, the Mathematics are Friends to Religion; inasmuch as they charm the Passions, restrain the Impetuosity 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, Besides, MathematicalStudies may serve for a pleasant Entertainment for those Hours, which young Men are apt to throw away on their Vices; the Delightfulness of them being such, as to make Solitude not only easy, but desirable.
What I have said may serve to recommend Mathematics for acquiring a vigorous 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 cast Extent and Usefulness in other Parts of Knowledge, And here it might suffice to tell you, that Mathematics is the Science of Quantity, or the Art of Reasoning about things that are capable of More and Less; and that the most Part of Objects of our Knowledge are such; as Matter, Space, Number, Time, Motion, Gravity, &c. We have but imperfect Ideas of Things without Quantity, and as imperfect a one of Quantity itself without the Help of Mathematics. All the visible Works of God Almighty are made in Number, Weight, and Measure: Therefore, to consider them, we ought to understand Arithmetic, Geometry, and Statics: And the greater Advances we make in those Arts, the more capable we are of considering such things, as are ordinary Objects of our Conceptions. But this will father appear from Particulars.
And, first, if we consider, to what Perfection we now know the Course, Periods, Order, Distances, and Proportions of the several great Bodies of the Universe, at least, such as fall within our View; we shall have Cause to admire the Sagacity and Industry [p.9] of the Mathematicians; and the Power of Numbers and Geometry well applied. 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 Knowledge of the Kalendar, and the Periods of the Seasons, without Instruments to make Observations, or any the least Notion of Observations, or any the least Notion of Observations or Instruments. When is it 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 Inspection of the Heavens, to find hoe 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 b e and is daily done, by what is known in Astronomy; yet when I consider the vast Industry, sagacity, Multitude of Observations, and other extrinsic 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 to let so much into the Knowledge of the Machine of the Universe, and Motion of its Parts, by the Rules of this Science, perhaps the Invention may seem easy. But when we reflect, 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 Pliny say, first predicted Eclipses; and his Scholar Anaximander Milesius, who found out the globous Figure of the Earth, the Equinoctial Points, the Obliquity of the Ecliptic, the Principles of Gnomonics, and made the first Sphere [p.10] or Image of the Heavens; and Pythagorus, to whom we owe the Discovery of the true System of the World, and Order of the Planets: Though, it may be, they were assisted by the Egyptians and Chaldeans. But whoever they were, that first made these Steps in this noble Art, they deserve the Praise and Admiration of all future Ages.
Inque domos superas scandere cura fuit!
Credibile est illos pariter vitiisque locisque
Altius humanis exseruisse caput.
Non Venus & vinum sublimia pectora fregit;
Officiumve fori, militiaeve labor.
Nec levis ambitio, perfusaque gloria fuco,
Magnarumve fames sollicitavit opum.
Admovere oculis distantia sidera nostris;
AEtheraque ingenio supposuere suo.
Ovid. in Io Fast.
But tho' the Industry of former Ages had discover'd the Periods of the great 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 Nation, the grand Secret of the whole Machine; which, now it is discovered, proves to be (like the other Contrivances of Infinite Wisdom) simple and natural, depending on 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 completes its Period in less than half the Age of Man, but likewise of the Comets, some of which, it is [p.11] 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 Procession of the Equinoctial Points, the Tides, the unequal Vibration of pendulous Bodies in different Latitudes, &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 Tinker durst hardly have hoped for; and, unless Mankind turn barbarous, will continue the Reputation of this Nation, as long as the Fabric of Nature shall endure. After this, what is it we may not expect from Geometry, joined to Observations and Experiments?
The next considerable Object of Natural Knowledge, I take to be Light. How unsuccessful Inquiries 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 Philosophers; whom nothing will serve, forsooth, 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 Inquiries about the Nature of Light, have discovered Two remarkable Properties of it, in the Reflexion and Refraction of its Beams: And from those, and their Streightness in other Cases, have invented the noble Arts of Optics, Catoprics, and Doptrics; teaching us to manage this subtile Body for the Improvement of our Knowledge, and useful Purposes of Life. They have likewise demonstrated the Causes if several celestial Appearances, that arise from the Inflexion of its Beams, both in the heavenly Bodies themselves, and other Phaenomena, as Parhelia, the Iris, &c. and by a late Experiment they have [p.12] discovered the Celerity of its Motion. And we shall know yet more surprizing Properties of Light, when Mr. Newton shall be pleased to gratify 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 presents to out View: And, I think, we know little of either, but what is owing to Mechanics and Geometry. The Two chiefest Properties of Air, its Gravity, and elastic 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 Torricellian Experiment. From this likewise, by Assistance of Geometry, they have determined the Height of the Atmosphere, as far as it has any sensible Density; which agrees exactly with another Observation of the Duration of Twilight. Air and Water make up the Object of the Hydrostatics, tho' denominated only from the latter, of which the Principles were long since settled and demonstrated by Archimedes, in his Book ωερι τωυ Оχγμέυωυ where are demonstrated the Causes of several surprising Phaenomena of Nature, depending only on the Aequlibrium of Fluids, the relative Gravities of these Fluids, and of Solids swimming or sinking therein. Here also the Mathematicians consider the different Pressures, Resistances, and Celerities of Solids moved in Fluids: From whence they explain a great many Appearances of Nature, unintelligble to those who are ignorant of Geometry.
Next, if we descend to the Animal Kingdom, there we may see the brightest Strokes pf Divine Mechanics. And whether we consider first the Animal Oeconomy in general, either in the internal Motion and Circulation of the Juices forced through the [p.13] several Canals by the Motion of the Heart, or their external Motions, and the instruments wherewith these are performed, we must reduce them to Mechanical Rules, and confess the Necessity of the Knowledge of Mechanics to understand them, or explain them to others. Borelli in his excellent Treatise de Motu Animalium, Steno in his admirable Myologiae Specimen, and other Mathematical Men, on the one hand, and the nonsensical, unintelligible Stuff that the common Writers on these Subjects have filled their Books with, on the other, are sufficient Instances to shew, how necessary Geometry is in such Speculations. The only Organ of an Animal Body, whose Structure and Matter of Operation is fully understood, has been the only one which the Geometers have taken to their Share to consider. It is incredible, how silly 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 applied themselves particularly to this, and valued themselves on it: And Galen pretended a particular Divine Commission to treat of it, Nay, notwithstanding the full Discovery of it, some go on in copying their Predecessors, and talk as Ungeometrically as ever. It is true, we cannot reason so clearly of the internal Motions of an Animal Body, as of the external, wanting sufficient Data, and decisive Experiments: But what relates to the latter (as the Articulation, Structure, Insertion, and Vires of the Muscles) is as subject to strict Mathematical Disquisition, 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 antient Physicians, that Mathematics are necessary for the study of Medicine itself, for which I could bring long Quotations out [p.14] of their Works. Among the Letter that are ascrib'd to Hippocrates, there is one to his Son Thessalus, recommending to him the Study of Arithmetic and Geometry, as necessary to Medicine. Galen in his Book, intituled, οτι αειςος ιατρος χαι ΦιλόσοφQ, begins
If one of the Reasons of the Antients 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 Influence upon Men and their Diseases, (and positively to deny it, and say, that they have not at all, is the Effect of Want of Observation) we have a much better and undoubted one in its room; viz. That Mathematics are found to be the best Instrument of promoting Natural Knowledge. 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 still the greater Need of Geometry. It is a pity, that the Qualities of an expert Anatomist, 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 employ his Time, and reward his Industry.
[p.15] As for the other two Kingdoms; Borelli and other Mathematical Men, seem to have talked very clearly of Vegetation: And Steno, another Mathematician in his excellent Treatise de Solido intra Solidum naturaliter conento, has apply'd this Part of Learning very handsomely to Fossils, and some other Parts of Natural History. I shall add only one thing more, That if we consider Motion itself, the great Instrument of the Actions of Bodies upon one another, the Theory of it is intirely owing to the Geometers; who have demonstrated its Laws both in hard and elastic Bodies; shewed how to measure its Quantity, how to compound and resolve 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 Knowledge, in considering several Motions that happen upon the Earth; viz. As to the Descent of heavy Bodies; The Curve of Projectiles; The Descent and Weight of heavy Bodies, when they lie on inclined Planes; The Theory of the Motion of Pendulous Bodies, &c.
From what I have said, I shall draw but one Corollary, That a Natural Philosopher without Mathematics is a very odd Sort of Person, that reasons about things that have Bulk, Figure, Motion, Number, Weight, &c. without Arithmetic, Geometry, Mechanics, Statics, &c. I must needs say, I have the last Contempt 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 pedibus illotis ad Philosophos divertunt, non hoc est satis, quod sint omnino, αθεώρητοι, αμγσοι, αγεωμετρηλοι sed legem etiam dant, quȃ Philosophari discant.
The Usefulness of Mathematics in several other Arts and Sciences is fully as plain. They were looked [p.16] upon by the antient Philosophers as to the Key to all Knowledge. Therefore Plato wrote upon his School, Ουδείς αγεωμέτρητος εισείτω, Let none unskilled in Geometry enter; and Xenocrates told one ignorant in Mathematics, who desired to be his Scholar, that he was fitter to card Wool, λαζας γαξ γχ εχες Φιλοσοφίας, You want the Handle of Philosophy, viz. Geometry. There is no understanding the Works of the antient Philosophers without it. Theo Smyrnaeus has wrote a Book, intituled, An Explanation of those things in Mathematics, that are necessary for the Reading of Plato: Aristotle illustrates his Precepts, and other Thoughts, by Mathematical Examples; and that not only in Logic, &c. but even in Ethics, where he makes use of Geometrical and Arithmetical Proportion, to explain commutative and distributive 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, without 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 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 Reader's Notion of the very Action itself. Just so, it is necessary to know the technical or doctrinal Part of Chronology, if a Man would be thoroughly skilled in History, it being impossible, without it, to unravel the Confusion of Historians. I remember Mr. Halley has determined [p.17] the Day and Hour of Julius Caesar'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 the Eclipses, 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 some otherwise Eminent have blundered, when they meddled with things that relate to these, and border on them, is too apparent.
Nobody, I think, will question the Interest, that Mathematics have in Painting, Music, and Architecture, which are all founded on Numbers, Perspective and the Rules of Light and Shadows, are owing to Geometry and Optics: And, I think, those Two comprehend pretty near the whole Art of Painting, except Decorum and Ordinance; which are only due Observance of the History and Circumstances 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, Landskip, 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 Perspective of the former, to the ordinary established Rules.
If Mathematics had not reduced Music to a regular System, by contriving its Scales, it had been no Art, but enthusiastic Rapture, left to the roving fancy of every Practitioner. This appears by the extraordinary Pains, which the Antients have taken to fit Numbers to Three Sorts of Music, the Diatonic, [p.18] Chromatic, and Enharmonic: Which if we consider with their Nicety in distinguishing their several Modes, we shall be apt to judge, they had something very fine in their Music, at least, for moving the Passions with single Instruments and Voices. But Music had been imperfect still, had not Arithmetic 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 Antients, to which we owe all the regular and noble Harmony of our modern Music.
As for Civil Architecture (of Military I shall speak afterwards) there is hardly any Part of Mathematics, but is some way subservient to it, Geometry and Arithmetic, 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 both firm and beautiful: Mechanics, for its Strength and Firmness, transporting and raising Materials: And Optics, 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 Vitruvius requires these, and many more, for making a complete Architect. I must own, that should any one set up to practise in any of the fore-mentioned Arts, furnished only with his Mathematical Rules, he would produce but very clumsy Pieces. He, that should pretend to draw by the Geometerical Rules of Perspective, or compose Music merely 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 Mathematics, as laying the Foundation of their Theory, and affording them Precepts, which, being once invented, are securely rely'd [p.19] upon by Practitioners. Thus many design, that know not a Tittle of the Reason of the Rules they practice by; and many, no better, perhaps, than he could have done, that invented the Scale, and the Numbers upon which their Harmony is founded. As Mathematics laid the Foundation of these Arts, so they must improve them: And he, that would invent, must be skilled 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 immediate Usefulness of Mathematics in Civil Affairs. To begin with Arithmetic, it were an endless Task to relate its several Uses in public and private Business. The Regulation and quick Dispatch pf both seen intirely owing to it, The Nations, that want it, are altogether barbarous, as some Americans, who can hardly reckon above Twenty. And, I believe, it would go bear to ruin the Trade of the Nation, were the easy Practice of Arithmetic abolished: For Example, were the Merchants and Tradesmen obliged to use of no other than the Roman way of Notation by Letters, instead of our present. And if we should feel the Want of our Arithmetic in the earliest Calculations, how much more in those, that are something harder? as Interest simple and compound, Annuities, &c. in which, it is incredible, how much the ordinary Rules and Tables influence the Dispatch of Business. Arithmetic is not only the great Instrument of private Commerce, but by it are (or ought to be) kept the public Accounts of a Nation: I mean those, that regard the whole State of a Commonwealth, as to the Number, Fructification of its People, Increase of Stock, Improvement of Lands and Manufactures, Balance of [p.20] Trade, Public Revenues, Coinage, Military Power by Sea and Land, &c. Those that would judge or reason truly about the State of any Nation, must go that way to work, subjecting all the fore-mentioned Particulars to Calculation. This is the true Political Knowledge. In this respect the Affairs of a Commonwealth 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 Countrymen, have wrote in Political Arithmetic, does abundantly shew the Pleasure and usefulness of such Speculations. It is true, for want of good Information, their Calculations sometimes 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 Command of all public Records?
Lastly, Numbers are applicable even to such as depend on Chance; the Quantity of Probability and Proportion of it in any Two proposed cases being subject to Calculation as much as anything else. Upon this depend the Principles of Game. We find Sharpers know enough of this, to cheat some Men that would take it very ill to be thought Bubbles: And one Gamester exceeds another, as he had a greater Sagacity and Readiness in calculating his Probability to win or lose in any proposed Case. To understand the Theory of Chance thoroughly, requires a Knowledge of Numbers and a pretty competent one of Algebra.
The several Uses of Geometry are not much fewer than those of Arithmetic. It is necessary for ascertaining of Property both in Planes and Solids, or in Surveying and Gauging. By it, Land is sold by the Measure, as well as Cloth: Workmen are paid [p.21] the due Price of their Labour, according to 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 preventing of Frauds and Controversies. I need not mention the measuring Distances, 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 applied: Of its Use in Civil, Military, and Naval Architecture, we shall speak afterwards.
From Astronomy we have the regular Disposition of our Time, in a due Succession of Years, which are kept within their Limits as to the Return of the Seasons, and the Motion of the Sun. This is no small Advantage for the due Repetition of the same Work, Labour, and Actions. For many of our Public, Private, Military, and Country Affairs, Appointments, &c. depending on the Products 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 undetermined 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 Celebration of the Church-Feasts and Fasts, according to the antient Custom, and primitive Institution; and, likewise, for the knowing of the Ebbing and Flowing of the Tides, the Spring, and Neap Tides, Currents, &c. So that whatever some People may think of an Almanac where all these are set down, it is oftentimes the most useful Paper that is published the same Year with it: Nay, the [p.22] 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 Disposition of the Year, and, at present, among the Turks, who have the same Confusion in their History as in their Kalendar. Therefore, a Matter of such Importance might well deserve the Care of the Great Emperor, 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,
Stellarum, coelique plagis, superisque vacavi.
The Mechanics have produced so many useful Engines, subservient to Conveniency, that it would be a Talk too great to relate the several Sorts of them: Some of them keep Life itself from being a Burden. If we consider such, as are invented for raising Weights, and are employed in Building, and other great Works, in which no Impediment is too great for them; or Hydraulic Engines for raising of Water, serving for great Use and Comfort to Mankind. where they have no other way to be supply'd readily with that necessary Element; or such as, by making Wind and Water work for us, save animal Force, and great Charges, and perform 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 Necessaries; 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 extol so excellent an Art. What shall we say [p.23] of the several Instruments, which are contrived to measure Time? We should quickly find the Value of them if we were reduced to the Condition of those barbarous Nations that want them. The Pendulum Clock, invented and completed by that famous Mathematician Monsieur Huygens, is an useful Invention. Is there any thing more wonderful, than several Planetary Machines, which have been invented to shew the Motions of the heavenly Bodies, and their Places at any time? Of which the most ingenious, according to the exactest Numbers, and true System, was made by the same M. Huygens: To which we may very justly apply Claudian's noble Verses upon that of Archimedes:
Risit, & ad superos talia dicta dedit:
Huccine mortalis progressa potentia curae?
Jam meus in fragili luditur orbe labr.
Jura poli, rerumque fidem, legesque Deorum
Ecce Syracusius transtulit arte senex.
Inclusus variis famulatur spiritus astris,
Et vivum certis motibus urget opus.
Percurrit proprium menititus Signifer annum,
Et simulata novo Cynthia mense redit.
Jamq; suum volvens audax industria mundum
Gaudet, & humanȃ sidera mente regit.
Quid falso insontem tonitru Salmonea miror?
AEmula naturae parva reperta manus.
Here I ought to mention the Sciatherical 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 Phillipus, their Censor, set up a better; which, no doubt, at that time, was thought a [p.24] 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 Street. To this Sort of Engines ought to be referred Spheres, Globes, Astrolabes, Projections 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 Motions, and to visit the most distant Places of the Earth, without the Fatigue and Danger of Voyages; to determine concerning 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 Inhabitants. To all these I might add those Instruments, which the Mathematicians have invented to execute their own Precepts, for making Observations either by Sea or Land, Surveying, Gauging, &c.
The Catoptrics and Doptrics furnish us with Variety of useful Inventions, both for promoting of Knowledge, and the Conveniencies of Life; whereby Sight, the great Instrument of our Perception, is so much improved, that neither 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 it is apply'd to here below, as the descrying Ships, and Men, and Armies, at a Distance, we have, by its means, discovered new Parts of the Creation, fresh Instances of the surprising Wisdom of the adorable Creator. We have, by it, discovered the Satellites of Jupiter, the Satellites and Ring of Saturn, the Rotation of the Planets about their own Axes; besides other Appearances, whereby the System of the World is made plain to Sense, as was before to Reason. The Telescope has also improved the Manner of Astronomical Observations, and made them much more accurate, than it was possible for them [p.25] to be before. And these Improvements in Astronomy, have brought along with them (as ever) correspondent Improvements in Geography. From the Observation of Jupiter's Satellites. 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 Minuteness escape our naked Eye. By it Men have pursued Nature into its most retired Recesses; so that now it can hardly any more hide its greatest Mysteries 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 Sciences had never gone further, than by their single Specula and Lentes to give those surprising 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 greatest Esteem, and invited to the closest Study: Especially, if we consider, that such as naturally are almost blind, and either know not their nearest Acquaintance at the Distance of a Room's Breadth, or cannot read, in order to pass their time pleasantly, are, by Glasses adapted to the Defect of their Eye, set on a Level again with those that enjoy their Eye-sight best, and that without Danger,m Pain, or Charge.
Again, Mathematics are highly serviceable to a Nation in Military Affairs. I believe this will be readily acknowledged by every Body. The Affairs of War take in Number, Space, Force, Distance, Time, &c. (Things of Mathematical Consideration) in all its Parts, in Tactics, Castramentation, Fortifying, Attacking, and Defending. The Antients had more Occasion for Mechanics in the Art of War than we [p.26] have; Gunpowder readily producing a Force far exceeding all the Engines, they had contriv'd for Battery. And this, I reckon, has lost us a good Occasion of improving Mechanics; the Cunning of Mankind never exerting itself so much, as in their Arts of destroying one another. But, as Gunpowder has made Mechanics less serviceable to War; it has made Geometry more necessary: There being a Force of Resistance in the due Measures and Proportions of 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 attained its utmost perfection. And tho', where the Ground is regular, it admits but of small variety, the Measures being pretty well determined by Geometry and Experience, yet where the Ground is made up of natural Strengths and Weaknesses, it affords some Scope for Thinking and Contrivance. But there is another much harder Piece of Geometry, which Gunpowder has given us Occasion to improve, and that is the Doctrine of Projectiles; whereon the Art of Geometry 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 Tactics and Castramentation, Mathematics retain the same Place in them as ever. And some tolerable Skill in these is necessary for Officers, as well as for Engineers. An Officer, that understands Fortification, will, caeteris paribus, much better defend his Port, as knowing, wherein its Strengths consists, or make use of his Advantage to his Enemy's Ruin, than he that does not. He knows, when he leads never so small a party, what his Advantages and Disadvantages in Defending and Attacking are, how to make the best of his Ground, &c. And hereby can do truly more Service than another of as much [p.27] Courage, who, for want of such Knowledge, it may be, throws away himself, and a Number of brave Fellows under his Command; and it is well, if the Mischief reaches no further. As for a competent Skill in Numbers, it is so necessary to Officers, 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 paribus, Officers are not advanced in Proportion to their Skill in Mathematical Learning; except for sometimes 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 Learning 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 studied, and merit the greatest Encouragement from a Nation, that owes to it both its Riches and Security. And not only doe the common Art of Navigation depend on Mathematics, but whatever Improvements shall be made in the Architecture Navalis, or Building of Ships, whether they are designed for Merchant-Ships, or Ships of War, whether swift running, or bearing a great Sail, or lying neat the Wind, be desired, these must all be the Improvements of geometry. Ship-Carpenters, indeed, are very industrious; but in these things they acknowledge their Inability, confess that their best Production are the Effects of Chance, and implore the Geometer's Help. Nor will common Geometry do the Business; it requires the most abstruse to determine the different Sections of [p.28] a Ship, according as it is designed for any of the aforesaid Ends, A French Mathematician P. le Hoste has lately endeavoured something in this way: And tho' it is not free from Errors, as requiring a fuller Knowledge of Geometry; yet is the Author much to be commended for this, as having bravely designed, 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 Tactics, or the regular Disposition of a Fleet in Attacking, Fighting, and Retreating, according to the different Circumstances of Wind, Tides, &c.
The great Objection, that is made against the Necessity of Mathematics, in the fore-mentioned great affairs of Navigation, the Art Military, &c. is, that we see those Affairs are carried on and managed by such, as are not great Mathematicians; as Seaman, Engineers, Surveyors, Gaugers, Clock-makers, Glass-grinders, &c. and that the Mathematicians are commonly speculative retired, 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 Mathematicians have not only invented and ordered all the Arts above-mentioned, by which those grand Affairs are managed; but have laid down Precepts, contrived Instruments and Abridgements so plainly, that common Artificers are capable of practicing 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 Affairs demand so great a Number of People to manage them, that it is impossible to breed so many good or even tolerable Mathematicians. The only thing then to be done was to make their Precepts so plain, that they might [p.29] be understood and practiced by a Multitude of Men. This will best appear by Examples. Nothing is more ordinary than Dispatch of Business by common Arithmetic, by the Tables of simple and compound Interest, Annuities, &c. Yet how few Men of Business understand the Reasons of common Arithmetic, or the Contrivance 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 top breed 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 Frustrums of Conoids and Spheroids, in every Market-Town, where such Work is necessary: The Mathematicians have therefore incrib'd such Lines on their common Rulers, and Slipping Rulers, and adapted so plain Precepts for them, that every Country-Carpenter, and Gauger, can do the Business accurately enough; though 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 contrived 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 Triangles every Day, that know not the Reason of any one of their Operations Seamen in their [p.30] Calculations make use of artificial Numbers, or Logarithms, that know nothing of their Contrivance: And indeed all those great Inventions of the most famous Mathematicians had been almost useless for those common and great Affairs, had not the Practice of them been easy to those who cannot understand them. From thence it is plain, that is it those Speculative 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 Conveniences of Life. They were the Men, that taught the World to apply Arithmetic, Astronomy, and Geometry, to Sailing, without which the Needle would be still useless. Just the same way in the other parts of Mathematics, the Precepts that are practiced by Multitudes, without being understood, were contriv'd by some few great Mathematicians.
Since then, it has been shewn, how much Mathematics improve the Mind, how subservient they are to other Arts, and how immediately useful to the Commonwealth, there needs no other Arguments or Motives to a Government, to encourage them, This is the natural Conclusion from the Premises. Plato, in his Republic, (lib. VII.) takes care, That, whoever is to be educated for Magistracy, or any considerable Post in the Commonwealth, may be instructed first in Arithmetic, then in Geometry, and thirdly in Astronomy. And however necessary those Arts were in Plato's time, they are much more so now: The Arts of War and Trade requiring much more the Assistance of those Sciences now, than did then: as being brought to a greater Height and Perfection. And accordingly we see, these Science are the particular Care of Princes, that design to raise the [p.31] Force and Power of their Countries. It is well known, that this is none of the least Arts, whereby the French King has brought his Subjects 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 Designing, 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 Seamen, where they must have Charts, Globes, Spheres, Compasses, Quadrants, Astrolabes, and all Books and Instruments necessary to teach their Art. The Directors of Hospitals are obliged to send thither yearly two or three of their Boys to be taught, and to furnish them with Books to be taught, and to furnish them with Books and Instruments. Those Professors are oblig'd to examine the journals deposited in the Office of the Admiralty, in the Place of their Establishment; to correct the Errors in Presence of the Seamen, and to restore them within a Month.' &c. King Charles the Second, who well understood the importance of Establishments of this Nature, founded one such school in Christ's Hospital, London; which, I believe, is inferior 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, established 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 restoring and continuing this, even in Peace, be note expedient for the breeding [p.31] of Engineers who are found so useful and valuable, and so difficult to be had in Time of War, and so little dangerous in Times of Peace.
Besides the Crowd of Merchants, Seamen, Surveyors, Engineers, Ship-Carpenters, Artisans, &c. that are to be instructed in the Practice of such Parts of Mathematics, as are necessary to their own business respectively, a competent Number 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, cannot overcome. And no doubt it is no small Impediment to the Advancement of Arts, that Speculative Men, and good Mathematicians, are unacquainted with their particular Defects, and the several Circumstances in them, that render things practicable of impracticable. But if there were public Encouragement, we should have skillful Mathematicians employed in those Arts, who would certainly find out and remedy the Imperfections of them. The present Lords Commissioners of the Admiralty, knowing that there are still two great Desiderata in Navigation, to wit, The Theory of the Variation of the magnetical Needle and a Method of finding out the Longitude of any Place, that may be practicable at Sea by Seamen, and being sensible, of what Importance it would be to find out either of them, have employed a very fit Person, the ingenious Mr. Halley, who has joined an intire Acquaintance in the Practice, to a full and thorough knowledge of the more abstruse Parts of Mathematics. And now that he is returned, 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 [p.33] 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 the Coasts where he touched, and by them others whose Relation to the former is known, the Nation is more than triply paid: 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.
The next thing that is necessary for the Improvement of Mathematical Learning, is, That Mathematics be more generally studied at our Universities 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 many be Comrades in Study; and the Direction of the Professors. There are also in Perfection all the Incitements to this Study; and especially an Acquaintance with the Works of the Antients, where this Learning is much recommended. There other Faculties are studied, 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 public Employments, where Mathematical Learning is absolutely necessary, and, without which, they, tho' of great natural Parts, must be at the Mercy and Discretion of their Servants and Deputies; [p.34] who will first cheat them, and then laugh at them. And not only public Employments, but their private Concerns, 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 relied 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
&c.) but in particular, it is best Companion for Country Life. Were this once become a fashionable Study, (and the Modeexercises its Empire over Learning as well as other things) it is hard to tell, how far it might influence the Morals of our Nobility and Gentry, in rendering them the more serious, diligent, curious; taking them off from the more fruitless and airy Exercises of the Fancy, which they are apt to run into.
The only Objection I can think of, that is brought against these Studies, is, That Mathematics require a particular 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 imagine, that a Man must be born a Mathematician. I answer, That this Exception is common to Mathematics and other Arts. That there are Persons that have a particular 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 Mathematics, than the others. A Man of good Sense and Application is the Person that [p.35] 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 Direction the Defect may be supplied, 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 general concerning the Order and Method of studying Mathematics; which I shall do very briefly, as knowing that you are already acquainted with the best Methods; and others with you may have them easily from the best and ablest Hands.
First, then, I lay down for a Principle, That nobody at an University is to be taught the Practice of any Rule without the true and solid Reason and Demonstration of the same. Rules without Demonstration must and ought to be taught to Seamen, 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 University, 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 Defects of the Arts; and therefore nothing must come, who are able to remedy the Defects of the Arts; and therefore nothing must be taken on Trust there. Seamen and Surveyors, &c. remember their Rules, because they are perpetually practicing them; but Scholars, who are not thus employ'd, if they know not the Demonstration of them, presently forget them.
Secondly, No Part of Mathematics ought to be taught by Compendiums. This follows from the former. Compendiums are fit to give a general and superficial Knowledge, not a thorough one. It is Time, and not the Bulk of Books, we ought to be sparing of: And I appeal to any Person of [p.36] Experience, whether solid Knowledge is not acquir'd in shorter time by Books treating fully of their Subjects, than by Compediums and Abridgements,
From hence it follows, that the Elements of Arithmetic 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 Arithmetic, as demonstrating the Rules from the Operations themselves. There remain then the first Six Books for the Geometry of Planes, and the last Three for Stereometry. The rest ought to be read in their own Place, for the Perfection of Arithmetic. 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 Accompts and Affairs for Examples of the Operations of Arithmetic, to one that is afterwards to have a Concern that way; whereas, to a Man of the first Quality, Examples from the Increase and Decrease of the People, or from Land or Sea Force, and from the Tactics, ought to be proposed. For, it is certain, nothing makes one tir'd sooner, than the frivolous and trifling Examples, that are commonly brought for the Exercise of the Rules of Arithmetic and Geometry; tho' this is common to them with the other Arts, as Grammar, Logic, &c.
The Manner of Writing of the Mathematicians of This and the former Age makes Trigonometry, with the Manner of Constructing Tables, &c. almost Elementary: And the Practical Geometry, 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 Spherics which are perfectly well-handled by Theodosius, a full Insight into the Principles of astronomy will be necessary.
[p.37] Mechanics come next to be read, which are the Ground of a great Part of natural Learning; and, afterwards, Optics, Catoptrics, and Dioptrics.
But none of these, except the Elements, can be fully understood, until one is pretty well skill'd in Conic Sections: And all these are made more easy by some tolerable Skill in Algebra, and its Application to Geometry.
These Foundations being laid, and one may, with great Ease, pursue the Study of the Mathematics, as his Occasions require; either in its abstract Parts, and the more recondite Geometry, and its Application to Natural Knowledge; or in Mechanics; by prosecuting the Statics, Hydrostatics, Ballistics, &c.: Or in Astronomy, by its Application to Geography, Navigation, Gnomonics, Astrolabes, &c. But, in most of these, a particular Order is not necessary: Any one may take That first, which he is most inclined to.
I shall not offer you any Advice concerning the Choice of Books; but refer you (if you want any) to the Direction of those who are eminent among you 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 with your Studies, I am,
Your Friend and Servant.