AN ESSAY ON The
Usefulness
of MATHEMATICAL LEARNING.
AN ESSAY ON The
Usefulness
of MATHEMATICAL LEARNING, IN A Letter from a
Gentleman
in the CITY to his
Friend
in
OXFORD.
Printed at the THEATER in
Oxford
for
Anth. Peisley
Bookseller, 1701.
Imprimatur,
RO. MANDER,
Vice-Can.
Oxon.
January 28. 1700./1701.
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
Ʋ niversity.
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 improvement in them. I have here sent you some short reflections upon the
Ʋ sefulness
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
implies no less; by which they were called either for their excellency, or because of all the
Sciences
they were first taught, or because they were judg'd to comprehend
. And amongst those, that are commonly reckoned to be the seven Liberal 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 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 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 Genius 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 Nation in general, than by shewing,
That the Mathematicks of all parts of humane knowledge, for the improvement of the Mind, for their subserviency to other Arts, and their usefulness 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 Mathematicks, yet think themselves Masters of all valuable Learning: but their displeasure must not deterr 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. 3dly.
In freeing it from
prejudice, credulity,
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 obvious. 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 delight is the more pure, free from the 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 fatentur 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,
. Therefore he calls it
. And indeed Youth is generally so much more delighted with
Mathematical
studies, than with the unpleasant tasks, that are some times imposed upon them, that I have known some reclaimed by them from idleness and neglect of 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 contriv'd by Nature to learn by Imitation more than by Precept: And I believe in that respect Reasoning is much like other inferiour Arts (as
Dancing, Singing,
&c.) acquired by practice. By accustoming our selves to Reason closely about quantity, we acquire a habit of doing so in other things. It is surprizing to see, what superficial, inconsequential Reasonings, satisfie 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 effeminacy of Mankind in being perswaded where they are delighted, have made them the sport of Orators, Poets, and Men of wit. Those
lumina Orationis
are indeed very good diversion 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. Logical precepts are more useful, nay, they are absolutely necessary for a rule of formal arguing in publick disputations, and confounding an obstinate and perverse adversary, and exposing him to the audience, or readers. But in the search of truth, an imitation of the method 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 imitate in the regular disposition and gradual progress of our enquiries; 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 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 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 the several 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 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
Mathematicks
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 perswaded 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 perswade to the contrary Opinion. Perhaps, 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 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,
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 recommend
Mathematicks
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 vast extent and Usefulness in other parts of knowledge. And here it might suffice to tell you, that
Mathematicks
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 the 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 it self without the help of
Mathematicks.
All the visible works of God Almighty are made in
number, weight,
and
measure;
therefore to consider them, we ought to understand
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 ordinary objects of our Conceptions. But this will farther appear from particulars.
And first, if we consider, to what perfection we now know the Courses, 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 of the
Mathematicians,
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 knowledge 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 inspection 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 consider the vast Industry, Sagacity, multitude of Observations, and other extrinsick 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 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 Aequinoctial Points, the Obliquity 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 Io . 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 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 upon the most known and most common property of matter, viz.
gravity.
From this the incomparable Mr .
Newton
has demonstrated the Theories of all the Bodies of the Solar System, of all the primary Planets and their secondaries, and among others, the Moon, which seem'd most averse to numbers: And not only of the Planets, the slowest of which compleats its Period in less than half the Age of a Man, but likewise of the Comets, 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 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 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 endure. After this, what is it, we may not expect from
Geometry
join'd to
Observations
and
Experiments?
The next considerable object of Natural 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
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 enquiries about the
Nature
of
Light,
have discovered two remarkable properties of it, in the reflection 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 several Coelestial appearances, that arise from the inflection of its Beams, both in the Heavenly Bodies themselves and other Phoenomena, as
Parhelia,
the
Iris,
&c. and by a late Experiment they have discovered the celerity of its motion. And we shall know yet more surprizing properties of
Light,
when Mr .
Newton
shall be pleas'd to gratifie the World with his
Book of Light and Colours.
The
Fluids
which involve our Earth, viz.
Air
and
Water,
are the next great and conspicuous Bodies, that Nature presents 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 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 the Twilight.
Air
and
Water
make up the object of the
Hydrostaticks,
tho' denominated only from the latter, of which the principles were long since settled and demonstrated by
Archimedes,
in his Book
, where are demonstrated the causes of several surprizing Phoenomena of Nature, depending only on the
Aequilibrium
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, 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 whither 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 Mechanical Rules, and confess the necessity 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 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 manner of operation is fully 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 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
Ʋ ngeometrically
as ever. It's 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 ancient Physicians, that Mathematicks are necessary 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
Geometry,
as necessary to Medicine.
Galen
in his Book intituled
, begins,
. 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 influence 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 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 a greater need of
Geometry.
It is 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.
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 contento,
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 it self, the great Instrument of the Actions of Bodies upon one another, the Theory of it is entirely owing to the
Geometers;
who have demonstrated its Laws both in hard and elastick Bodies; shew'd how to measure it's 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 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 motion of Pendulous Bodies, &c.
From what I have said, I shall draw but one Corollary, That a natural Philosopher without Mathematicks is a very odd sort of a person, that reasons about things that have
Bulk, Figure, Motion, Number, Weight,
&c. without
Arithmetick, Geometry, Mechanicks, Staticks,
&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 omninò
sed legem etiam dant, quâ Philosophari discant.
The usefulness of
Mathematicks
in several 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,
,
Let none unskilled in Geometry enter;
and
Xenocrates
told one ignorant in
Mathematicks,
who desired to be his Scholar, that he was fitter to Card Wooll,
,
you want the handle of Philosophy,
viz.
Geometry.
There is no understanding the works of the ancient Philosophers without it.
Theo Smyrnoeus
has wrote a Book entituled, An explanation of those things in Mathematicks, that are necessary for the reading of
Plato: Aristotle
illustrates his precepts and other thoughts by Mathematical examples, and that not only in
Logick,
&c. but even in
Ethicks,
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 Readers notion 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 remember Mr .
Hally
has determin'd the day and hour of
Julius Coesar's
Landing in
Britain,
from the circumstances of his relation. And every body knows, how great use our incomparable Historian Mr .
Dodwell
has made of the Calculated times of 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 blunder'd, when they meddled with things that relate to these, and border on them, is too apparent.
No body, I think, will question the interest, that Mathematicks have in
Painting, 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
decorum
and
ordinance;
which are only a 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
Mathematicks
had not reduced
Musick
to a regular System, by contriving its
Scales,
it had been no Art, but Enthusiastick Rapture, left to the roving fancy of every Practitioner. This appears by the extraordinary pains, which the Ancients have taken to fit numbers to three sorts of Musick, the
Diatonick, Chromatick,
and
Enharmonick:
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
Musick,
at least for moving the passions 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 Harmony 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, transporting 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
Vitruvius
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 Mathematical Rules, he would produce but very clumsy pieces. He, that should pretend to draw by the Geometrical Rules of Perspective, 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 affording 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 immediate usefulness of
Mathematicks
in
Civil Affairs.
To begin with
Arithmetick,
it were an endless task to relate its several 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 barbarous, as some
Americans,
who can hardly reckon above twenty. And I believe 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 ordinary 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 Nation: I mean those, that regard the whole State of a Common-wealth, as to the number, fructification of its people, increase 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-mentioned 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 Speculations. It is true, for want of good information, 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 command 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 proportion of it in any two proposed cases being subject to Calculation as much as any thing 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 has a greater sagacity and readiness in Calculating his probability to win or lose in any proposed case. To understand 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 property 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 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 apply'd: 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 Publick, 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 Ancient 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 what-ever some people 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 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 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 conveniency, 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 burden. 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 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 extoll so excellent an Art. What shall we say of the several Instruments, which are contriv'd 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 compleated by that famous
Mathematician
Monsieur
Hugens
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.
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
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 Philippus
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, 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 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 perception, is so much improv'd, 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 surprizing 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 it was before to
reason.
The
Telescope
has also improv'd the manner 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) 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 retir'd 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 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 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 rooms breadth, or cannot read in order to pass their time pleasantly, 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 serviceable 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
consideration) in all its parts, in
Tacticks, Castrametation, Fortifying, Attacquing,
and
Defending.
The Ancients had more occasion for Mechanicks in the Art of War than we have: Gun-powder 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 our Mechanicks: the cunning of Mankind never exerting it self so much, as in their Arts of destroying one another. But, as Gun-powder has made Mechanicks less serviceable to War; it has made
Geometry
more necessary: 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 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 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
Officer,
that understands Fortification, will
caeteris paribus
much better defend his post, as knowing, wherein its strength consists, 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 disadvantages 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 Courage, 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
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, 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 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 study'd, and merit the greatest encouragement from a Nation, that owes to it both its Riches and Security. And not only does the Common Art of
Navigation
depend on
Mathematicks,
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 improvements of
Geometry. Ship-Carpenters
indeed 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
Geometry
do the business; it requires the most abstruse to determine the different sections of a Ship, according as it is design'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
Geometry;
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 regular disposition of a Fleet in Attacquing, Fighting and Retreating, according to the different circumstances of Wind, Tides, &c.
The great objection, that is made against the necessity of
Mathematicks,
in the fore-mention'd great Affairs of
Navigation,
the Art
Military,
&c. is, that we see those Affairs are carry'd on and managed by such, as are not great Mathematicians; as Sea-men, Engineers, Surveyers, Gaugers, Clock-makers, Glassgrinders, &c. and that the Mathematicians 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 Mathematicians have not only invented and order'd all the Arts above-mentioned, by which those grand Affairs are managed; 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 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 be understood and practised by a multitude of Men. This will best appear by examples. Nothing is more ordinary than dispatch of business by common
Arithmetick,
by the
Tables of simple and compound Interest, Annuities
&c. Yet how few Men of business understand the reasons of common
Arithmetick,
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 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 inscrib'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
Logarithmes,
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 made easy to those who cannot understand them. From hence it is plain, that it is to 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 conveniencies of Life. They were the Men, that taught the World to apply
Arithmetick, Astronomy,
and
Geometry
to
Sailing,
without which the needle would be still useless. Just the same way in the other parts of
Mathematicks,
the Precepts that are practised by multitudes, without being 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 encourage them. This is the natural conclusion from these premises.
Plato
in his
Republick
(
lib.
VII.) takes care,
That, whoever 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 requiring much more the assistance of those
Sciences
now, than they did then; as being brought to a greater height and perfection. 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, 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 Sea-men, 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 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 deposited 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 understood the importance of Establishments 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, 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 not expedient for the breeding of Engineers, 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
Mathematicks,
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
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 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 Sea-men, and being sensible, of what importance it would be to find out either of them, have imployed a very fit person, the ingenious Mr .
Hally,
who has joyn'd an entire acquaintance in the practice, to a full and thorough knowledge of the more abstruse parts of
Mathematicks.
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 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 Learning,
is, That Mathematicks be more generally study'd at our
Ʋ niversities
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 Professors. There are also in perfection all the incitements to this study, and especially an acquaintance with the works of the Ancients, where this
Learning
is so much recommended: There other Faculties 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 Learning
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 Deputies; who will first cheat them, and then laugh at them. And not only Publick 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 rely'd upon without it. It not only makes a Man of Quality and Estate his whole Life more Illustrious, and more useful for all Affairs, (as
Hippocrates
says,
) but in particular, it is the best Companion for a Country Life. Were this once become a fashionable study (and the
Mode
exercises its 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 rendring them 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 Mathematicks 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 Mathematicks 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 Mathematicks than the others. A Man of good sense and application is the person, 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 direction 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 general concerning the
Order
and
Method
of studying
Mathematicks;
which I shall do very shortly, 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 no body at an
Ʋ niversity
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
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
Ʋ niversity,
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 be taken on trust there.
Sea-men
and
Surveyors,
&c. remember their Rules, because they are perpetually 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.
Compendiums
are fit to give a general and superficial 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 solid knowledge is not acquir'd in shorter time by Books treating fully of their subjects, than by Compendiums and Abridgements.
From hence it follows, that the
Elements
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 demonstrating 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
Arithmetick,
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 people,
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
Arithmetick
and
Geometry:
tho' this is common to them with the other Arts, as
Grammar, Logick,
&c.
The manner of Writing of the Mathematicians of this and the former Age makes
Trigonometry,
with the manner of constructing its 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
Sphericks,
which are perfectly well handled by
Theodosius,
a full insight into the principles of
Astronomy
will be necessary.
Mechanicks
come next to be read, which are the Ground of a great part of Natural Learning: and afterwards
Opticks, Catoptricks
and
Dioptricks.
But none of these except the Elements can be fully understood until one is pretty well skill'd in
Conick-sections:
And all these are made more easy by some tolerable skill in
Algebra,
and its application to
Geometry.
These foundations being laid, any one may with great ease pursue the study of the Mathematicks, as his occasions require: either in its abstract parts, and the more
recondite Geometry,
and its application to Natural knowledge; or in Mechanicks, by prosecuting the
Staticks, Hydrostaticks, Ballisticks,
or in
Astronomy,
by its application to
Geography, 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 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 in your studies, I am,
Sir,
Your Friend and Servant.
25. Novemb. 1700.
FINIS.