An essay on the usefulness of mathematical learning: in a letter from a gentleman in the city to his friend in Oxford.

AN ESSAY ON The Usefulness of MATHEMATICAL LEARNING, &c.

SIR,

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.

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 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 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 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 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

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

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

are furnished with Mathematical know|ledge, are prompt and quick at all other Sciences, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. Therefore he calls it 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 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.

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

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

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.

—Si propiùs stes, Te capiet magis.—

On the contrary, the Mathematicks are

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

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,

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

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.

Felices animae, quibus haec cognoscere primis, Inque domos superas scandere cura fuit. Credibile est illos pariter vitiisque jocisque Altius humanis exseruisse caput. Non Venus & vinum sublimia pectora fregit, Officiumque fori, militiaeque labor. Non levis ambitio, perfusaque gloria fuco, Magnarumque fames sollicitavit opum. Admovere oculis distantia sidera nostris, Aetheraque ingenio supposuere suo.Ovid. in I^o. Fast.

But tho' the industry of former Ages had discovered 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 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 M^r. 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,

&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

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 M^r. 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

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 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, 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

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

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 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, begins, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉

〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 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

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,

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 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 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

his School, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, 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, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, 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

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 M^r. 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 M^r. 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

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

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

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

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

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

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.

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,

&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

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,

—Media inter praelia semper Stellarum, Coelique plagis, superisque vacavi.

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

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.

Jupiter in parvo cum cerneret Aethera vitro, Risit, & ad superos talia dicta dedit: Huttine mortalis progressa potentia curae? Jam meus in fragili luditur or be labor. Jura poli, rerumque fidem, legesque Deorum Ecce Syracusais transtulit arte senex. Inclusus variis famulatur spiritus astris, Et vioum certis motibus urget opus. Percurrit proprium mentitus signifer annum, Et simulata novo Cynthia mense redit. Jam{que} suum volvens audax industria mundum Gaudet, & humanâ sidera mente regit. Quid falso insontem tonitru Salmonea miror? Aemula naturae parva reperta manus.

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

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

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

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,

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.

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

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

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

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

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:

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

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.

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

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.

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,

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 M^r. 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.

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;

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, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 &c;) 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.

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

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.

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

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.

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, &c; 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

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,

Sir,

Your Friend and Servant.

25. Novemb. 1700.

FINIS.