Authored
by Martin Strong, John
Arbuthnot
Third Edition, 1745.
[Note:
Greek phrases typed in the text below are approximate due to the
difficulty of matching available font characters, and
difficulty in reading the original.]
AN
ESSAY
ON THE
USEFULNESS
OF
Mathematical Learning
IN A
LETTER
FROM
A GENTLEMAN in the CITY to his FRIEND at OXFORD.
The THIRD EDITION.
LONDON:
Printed for J. BARRETT, Bookseller in Oxford; and Sold by S. BIRT, and
B. DOD,
Booksellers in Ave-Mary-Lane, near St. Paul's, London.
M.DCC.XLV.
[p.2]
Imprimatur
RO. MANDER,
Vice-Chanc. OXON.
Jan, 28,
1700,
1701.
[p.3]
SIR,
I AM glad to hear from you, that the Study of the Mathematics is
Promoted and Encouraged among the Youth
of your University.
The great Influence, which these Sciences
have on the Philosophy, and all useful Learning, as well as the
Concerns of the Public, may sufficiently recommend them to your Choice
and Consideration: And the particular Advantages, which You of that
Place enjoy, give Us just Reason to expect from You a suitable
Improvement in them. I have here sent you some short Reflections upon
the Usefulness
of Mathematical Learning,
which may serve as an argument to incite you to a closer and more
vigorous Pursuit of it.
In all Ages and Countries, where
Learning hath prevailed, the Mathematical
Sciences have been looked upon as the most considerable
Branch of it. The very Name Μάθησις implies
no less; by which they were
called, either for their Excellency; or because, of all the Sciences, they were
first taught; or because they were judged to comprehend
πάντα τα
Μαθήματε. [p.4] And,
amongst those,
that are commonly reckoned to be the Seven Liberal Arts, Four are
Mathematical, to wit, Arithmetic,
Music, Geometry, and Astronomy.
But, notwithstanding their Excellency
and Reputation, they have not been taught nor study'd to universally,
as some of the rest; which I take to have proceeded from the following
Causes: The Aversion of
the greatest Part of Mankind to serious Attention, and close Arguing; Their
not comprehending
sufficiently the Necessity, or great Usefulness, of these in other
Parts of Learning; An Opinion that this Study requires
a particular Genius and Turn of Head, which few are so happy as to be
born with; And the
Want of public Encouragement, and able Masters. For these,
and perhaps some other Reasons, this Study hath been generally
neglected, and regarded only by some Persons, whose happy Genius and
Curiosity have prompted then to it, or who have been forced upon it by
its immediate Subserviency to some particular Art of Office.
Therefore, I think I cannot do better
Service to Learning, Youth, and
the Nation in general, than by shewing, That the Mathematics, of all
Parts of human Knowledge, for the Improvement of the Mind, for their
Subserviency to other Arts, and their Usefulness to the Commonwealth,
deserve most to be encouraged. I know a Discourse of this
Nature will be offensive to some, who, while they are ignorant of
Mathematics, yet think themselves Masters of all valuable Learning: But
their Displeasure must not deter me from delivering an useful Truth.
The Advantages which accrue to the Mind
by Mathematical Studies, consist chiefly in these Things: 1st, In
accustoming it to Attention.
2dly, In giving it a Habit of close
and demonstrative
Reasoning. 3rdly, In freeing it from Prejudice, Credulity, and
Superstition.
[p.5]
First, the Mathematics make the Mind attentive to Objects, which it
considers. This they do by entertaining it with a great Variety of
Truths, which are delightful and evident, but not obvious. Truth is the
same thing to the Understanding, as Music to the Ear, and Beauty to the
Eye. The Pursuit of it does really as much gratify a natural Faculty
implanted in us by our Creator, as the pleasing of our Senses: Only in
the former Case, as the Object and Faculty are more spiritual, the
Delight is the more pure, free from Regret, Turpitude, Lassitude, and
Intemperance, that commonly attend sensual Pleasures. The most Part of
other Sciences consisting only of probable Reasonings, the Mind has not
where to fix; and, wanting sufficient Principles to pursue its Searches
upon, gives them over as impossible. Again, as in Mathematical Investigations Truth
may be found, so it is not always obvious:
This spurs the Mind, and makes it diligent and attentive. In Geometria (says Quinctilian, lib. I. cap. 10) partem satentur esse utilem
teneris aetatibus: agitari namque animos, atque acui ingenia, &
celeritatem percipiendi venire inde concedunt. And Plato (in Repub. lib. VII.)
observes, that the Youth, who are furnished with Mathematical Knowledge,
are prompt and quick at all other Sciences, ειςωάυλα
τα
Μαθήαλα
οξεις
φαίυουλαι.
Therefore, he
calls it χαλα
ωαιδείαυ
οδόυ. And, indeed, Youth
is generally so much more
delighted with Mathematical
Studies, than with the unpleasant Talks, that are sometimes imposed
upon them, that I have known some reclaimed by them from Idleness, and
Neglect of Learning; and acquire in time a Habit of Thinking,
Diligence, and Attention; Qualities, which we ought to study by all
means to beget in their desultory and roving Minds.
The Second Advantage, which the Mind
reaps from Mathematical
Knowledge, is a Habit of clear,
demonstrative, and methodical
Reasoning. We are [p.6]
contrived by Nature to learn by Imitation more than Precept: And, I
believe, in that respect, Reasoning is much like other inferior Arts
(as Dancing, Singing, &c.)
acquired by Practice. By accustoming
ourselves to reason closely about Quantity, we acquire a Habit of doing
so in other things. It is surprising to see, what superficial
inconsequential Reasonings satisfy the most Part of Mankind. A Piece of
Wit, a Jest, a Simile, or a Quotation of an Author, passes for a mighty
Argument: With such things as these are the most Part of Authors
stuffed; and from these weighty Premises they infer their Conclusions.
This Weakness and Effeminancy of Mankind in being persuaded where they
are delighted, have made the Sport of Orators, Poets, and Men of Wit.
Those lumina orationis
are indeed very good Diversion for the Fancy, but are of the Proper
Business of the Understanding; and where a Man pretends to write on
abstract Subjects in a scientifical Method, he ought not to debauch
them. Logical Precepts are more useful, nay, are absolutely necessary,
for a Rule of formal Arguing in public Dispositions, and confounding an
obstinate and perverse Adversary, and exposing him to the Audience of
Readers. But, in the Search of Truth, an imitation of the Method of Geometers will
carry a Man farther than all Dialectical
Rules. Their Analysis
is the proper Model we ought to form ourselves upon, and imitate in the
regular Disposition, and gradual Progress, of our Inquiries; and even
he, who is ignorant of the Nature of Mathematical Analysis,
uses a Method somewhat analogous to it. The Composition of the Geometers, or their
Method of demonstrating Truths already found out, viz. by Definitions of Words agreed
upon, by self-evident Truths, and propositions that have been already
demonstrated, is practicable in other Subjects, tho' not
to the same Perfection, the natural [p.7]
Want of Evidence in the things themselves not allowing it; but it is
imitable top a considerable Degree. I dare appeal to some Writings of
our own Age and Nation, the Authors of which have been mathematically
inclined. I shall add no more on this Head, but, that one, who is
accustomed to the methodical Systems of truths, which the Geometers have
reared up in several branches of those Sciences, but
endeavour, as far as he can, to reform them.
Thirdly, Mathematical
Knowledge adds a manly Vigour to the Mind, frees it from Prejudice, Credulity,
and Supersition.
This it does in two Ways: 1st, By accustoming us to examine, and not to
take things upon Trust. 2dly, By giving us a clear and extensive
Knowledge of the System of the World; which, as it creates in us the
most profound Reverence of the almighty and wise Creator; so it frees
us from the mean and narrow Thoughts, which Ignorance and Superstition
are apt to beget. How great an Enemy Mathematics are to
Superstition, appears from this, That in those Countries, where Romish Priests
exercise their barbarous Tyranny over the Minds of Men, Astronomers, who
are fully persuaded of the Motion of the Earth, dare not speak out: But
tho' the Inquisition
may extort a Recantation, the Pope, and a general Council too, will not
find themselves able to persuade to the contrary Opinion. Perhaps, this
may have given Occasion to calumnious Suggestion, as if Mathematics were an
Enemy to true Religion, which appears always to the best Advantage,
when it is most examined.
------ Si propiùs
stes,
Te capiet magis. ------
[p.8] On the
contrary, the Mathematics
are Friends to Religion; inasmuch as they charm the Passions, restrain
the Impetuosity of Imagination, and purge the Mind from Error and
Prejudice. Vice is Error, Confusion, and false Reasoning; and all Truth
is more or less opposite to it, Besides, Mathematical
Studies may serve for a pleasant Entertainment for those Hours, which
young Men are apt to throw away on their Vices; the Delightfulness of
them being such, as to make Solitude not only easy, but desirable.
What I have said may serve to recommend Mathematics for
acquiring a vigorous Constitution of Mind; for which Purpose they are
as useful, as Exercise is for procuring Health and Strength to the
Body. I proceed now to shew their cast Extent and Usefulness in other
Parts of Knowledge, And here it might suffice to tell you, that Mathematics is the Science of
Quantity, or the Art
of Reasoning about things that are capable of More and Less; and that the
most Part of Objects of our Knowledge are such; as Matter, Space,
Number, Time, Motion, Gravity, &c.
We have but imperfect Ideas
of Things without Quantity, and as imperfect a one of Quantity itself
without the Help of Mathematics.
All the visible Works of God Almighty are made in Number, Weight, and Measure:
Therefore, to consider them, we ought to understand Arithmetic, Geometry, and
Statics:
And the greater Advances we make in those Arts, the more capable we are
of considering such things, as are ordinary Objects of our Conceptions.
But this will father appear from Particulars.
And, first, if we consider, to what
Perfection we now know the Course, Periods, Order, Distances, and
Proportions of the several great Bodies of the Universe, at least, such
as fall within our View; we shall have Cause to admire the Sagacity and
Industry [p.9] of
the Mathematicians;
and the Power of Numbers
and Geometry
well applied. Let us cast our Eyes backward, and consider Astronomy in its
Infancy: Or rather let us suppose it still to begin: For instance; a
Colony of rude Country People, transplanted into an Island remote from
the Commerce of all Mankind, without so much as Knowledge of the
Kalendar, and the Periods of the Seasons, without Instruments to make
Observations, or any the least Notion of Observations, or any the least
Notion of Observations or Instruments. When is it we could expect any
of their Posterity should arrive at the Art of predicting an Eclipse?
Not only so, but the Art of reckoning all Eclipses that are past or to
come, for any Number of years? When is it we could suppose, that one of
those Islanders, transported to any other Place of the Earth, should be
able, by the Inspection of the Heavens, to find hoe he were South or
North, East or West of his own Island, and to conduct his ship back
thither? For my Part, tho' I know this may b e and is daily done, by
what is known in Astronomy;
yet when I consider the vast Industry, sagacity, Multitude of
Observations, and other extrinsic Things necessary for such a sublime
Piece of Knowledge, I should be apt to pronounce it impossible, and
never to be hoped for. Now we are to let so much into the Knowledge of
the Machine of the Universe, and Motion of its Parts, by the Rules of
this Science,
perhaps the Invention may seem easy. But when we reflect, what
Penetration and Contrivance were necessary to lay the Foundations of so
great and extensive an Art, we cannot but admire its first Inventors:
As Thales
Milesius, who, as Diogenes Laertius
and Pliny
say, first predicted Eclipses; and his Scholar Anaximander Milesius,
who found out the globous Figure of the Earth, the Equinoctial Points,
the Obliquity of the Ecliptic, the Principles of Gnomonics, and made
the first Sphere [p.10]
or Image of the Heavens; and
Pythagorus,
to whom we owe the Discovery of the true System of the World, and Order
of the Planets: Though, it may be, they were assisted by the Egyptians and Chaldeans.
But whoever they were, that first made these Steps in this noble Art,
they deserve the Praise and Admiration of all future Ages.
Felices animos, quibus haec
cognoscere primis,
Inque domos superas scandere cura fuit!
Credibile est illos pariter vitiisque locisque
Altius humanis exseruisse caput.
Non Venus & vinum sublimia pectora fregit;
Officiumve fori, militiaeve labor.
Nec levis ambitio, perfusaque gloria fuco,
Magnarumve fames sollicitavit opum.
Admovere oculis distantia sidera nostris;
AEtheraque ingenio supposuere suo.
Ovid. in Io
Fast.
But tho' the Industry of former Ages had discover'd the Periods of the
great Bodies of the Universe and the true System and Order of them, and
their Orbits, pretty near; yet was there one thing still reserved for
the Glory of this Age, and the Honour of the English
Nation, the grand Secret of the whole Machine; which, now it is
discovered, proves to be (like the other Contrivances of Infinite
Wisdom) simple and natural, depending on the most known and most common
Property of Matter, viz. Gravity.
From this the incomparable Mr. Newton
has demonstrated the Theories of all the Bodies of the Solar System, of
all the primary Planets, and their Secondaries, and, among others, the
Moon, which seem'd most averse to Numbers: And not only of the Planets,
the slowest of which completes its Period in less than half the Age of
Man, but likewise of the Comets, some of which, it is [p.11]
probable, spend more than 2000 Years in one Revolution about the Sun;
for whose Theory he has laid such a Foundation, that after Ages,
assisted with more Observations, may be able to calculate their
Returns. In a Word, the Procession of the Equinoctial Points, the
Tides, the unequal Vibration of pendulous Bodies in different
Latitudes, &c.
are no more a Question to those, that have Geometry enough to
understand what he has delivered on those subjects: A Perfection in Philosophy,
that the boldest Tinker durst hardly have hoped for; and, unless
Mankind turn barbarous, will continue the Reputation of this Nation, as
long as the Fabric of Nature shall endure. After this, what is it we
may not expect from Geometry,
joined to Observations
and Experiments?
The next considerable Object of Natural
Knowledge, I take to be Light.
How unsuccessful Inquiries are about this glorious Body without the
Help of Geometry,
may appear from the empty and frivolous Discourses and Disputations of
a Sort of Men, that call themselves Philosophers;
whom nothing will serve, forsooth, but the Knowledge of the very
Nature, and intimate Causes, of every thing: While, on the other hand,
the Geometers,
not troubling themselves with those fruitless Inquiries about the Nature of Light,
have discovered Two remarkable Properties of it, in the Reflexion and
Refraction of its Beams: And from those, and their Streightness in
other Cases, have invented the noble Arts of Optics, Catoprics,
and Doptrics;
teaching us to manage this subtile Body for the Improvement of our
Knowledge, and useful Purposes of Life. They have likewise demonstrated
the Causes if several celestial Appearances, that arise from the
Inflexion of its Beams, both in the heavenly Bodies themselves, and
other Phaenomena, as Parhelia,
the Iris, &c. and by
a late Experiment they have [p.12]
discovered the Celerity of its Motion. And we shall know yet more
surprizing Properties of Light,
when Mr. Newton
shall be pleased to gratify the World with his Book of Light and Colours.
The Fluids
which involve our Earth, viz. Air
and Water,
are the next great and conspicuous Bodies that Nature presents to out
View: And, I think, we know little of either, but what is owing to Mechanics and Geometry. The Two
chiefest Properties of Air,
its Gravity, and elastic Force, have been discovered by Mechanical
Experiments. From thence the Decrease of the Air's Density, according
to the Increase of the Distance of the Earth, has been demonstrated by Geometers, and
confirmed by Experiments of the Subsidence of the Mercury in the
Torricellian Experiment. From this likewise, by Assistance of Geometry,
they have determined the Height of the Atmosphere, as far as it has any
sensible Density; which agrees exactly with another Observation of the
Duration of Twilight. Air
and Water
make up the Object of the Hydrostatics,
tho' denominated only from the latter, of which the Principles were
long since settled and demonstrated by Archimedes, in his
Book ωερι
τωυ
Оχγμέυωυ
where are demonstrated the Causes of several surprising Phaenomena of
Nature, depending only on the Aequlibrium
of Fluids,
the relative Gravities of these Fluids,
and of Solids swimming or sinking therein. Here also the Mathematicians
consider the different Pressures, Resistances, and Celerities of Solids
moved in Fluids: From whence they explain a great many Appearances of
Nature, unintelligble to those who are ignorant of Geometry.
Next, if we descend to the Animal Kingdom,
there we may see the brightest Strokes pf Divine Mechanics. And whether
we consider first the Animal
Oeconomy in general, either in the internal Motion and
Circulation of the Juices forced through the [p.13]
several Canals by the Motion of the Heart, or their external Motions,
and the instruments wherewith these are performed, we must reduce them
to Mechanical Rules, and confess the Necessity of the
Knowledge
of Mechanics to understand them, or explain them to others. Borelli in his
excellent Treatise de
Motu Animalium, Steno
in his admirable Myologiae
Specimen,
and other Mathematical Men, on the one hand, and the nonsensical,
unintelligible Stuff that the common Writers on these Subjects have
filled their Books with, on the other, are sufficient Instances to
shew, how necessary Geometry
is in such Speculations. The only Organ of an Animal Body, whose
Structure and Matter of Operation is fully understood, has been the
only one which the Geometers
have taken to their Share to consider. It is incredible, how silly the
greatest and ablest Physicians talked of the Parts of the Eye and their
Use, and of the Modus
Visionis, before Kepler
by his Geometry
found it out, and put it past Dispute, tho' they applied themselves
particularly to this, and valued themselves on it: And Galen
pretended a particular Divine Commission to treat of it, Nay,
notwithstanding the full Discovery of it, some go on in copying their
Predecessors, and talk as Ungeometrically
as ever. It is true, we cannot reason so clearly of the internal
Motions of an Animal Body, as of the external, wanting sufficient Data, and decisive
Experiments: But what relates to the latter (as the Articulation,
Structure, Insertion, and Vires
of the Muscles) is as subject to strict Mathematical Disquisition, as
any thing whatsoever; and even in the Theory of Diseases, and their
Cures, those, who talk Mechanically talk most intelligibly. Which may
be the reason for the Opinion of the antient Physicians, that
Mathematics are necessary for the study of Medicine itself, for which I
could bring long Quotations out [p.14]
of their Works. Among the Letter that are ascrib'd to Hippocrates, there
is one to his Son Thessalus,
recommending to him the Study of Arithmetic
and Geometry,
as necessary to Medicine. Galen
in his Book, intituled, οτι
αειςος
ιατρος
χαι
ΦιλόσοφQ,
begins
If one of the Reasons
of the Antients for this be now somewhat unfashionable, to wit, because
they thought a Physician should be able to know the Situation and
Aspects of the Stars, which they believed had Influence upon Men and
their Diseases, (and positively to deny it, and say, that they have not
at all, is the Effect of Want of Observation) we have a much better and
undoubted one in its room; viz.
That Mathematics are found to be the best Instrument of promoting
Natural Knowledge. 2dly, If we consider, not only the Animal Oeconomy
in general, but likewise the wonderful Structure of the different Sorts
of Animals, according to the different Purposes for which they were
design'd, the various Elements they inhabit, the several Ways of
procuring their Nourishment, and propagating their Kind, the different
Enemies they have, and Accidents they are subject to, here is still the
greater Need of Geometry.
It is a pity, that the Qualities of an expert Anatomist, and
skillful Geometer,
have seldom met in the same Person. When such a one shall appear, there
is a whole Terra
incognita of delightful Knowledge to employ his Time, and
reward his Industry.
[p.15] As for the
other two Kingdoms; Borelli
and other Mathematical
Men, seem to have talked very clearly of Vegetation: And Steno, another
Mathematician in his excellent Treatise de Solido intra Solidum
naturaliter conento, has apply'd this Part of Learning
very handsomely to Fossils,
and some other Parts of Natural History. I shall add only one thing
more, That if we consider Motion itself, the great Instrument of the
Actions of Bodies upon one another, the Theory of it is intirely owing
to the Geometers; who
have
demonstrated its Laws both in hard and elastic Bodies; shewed how to
measure its Quantity, how to compound and resolve the several Forces,
by which Bodies are agitated, and to determine the Lines,
which those compound Forces make them describe: Of such Forces Gravity,
being the most constant and uniform, affords a great Variety of useful
Knowledge, in considering several Motions that happen upon the Earth; viz.
As to the Descent of heavy Bodies; The Curve of Projectiles; The
Descent and Weight of heavy Bodies, when they lie on inclined Planes;
The Theory of the Motion of Pendulous Bodies, &c.
From what I have said, I shall draw but
one
Corollary, That a Natural Philosopher without Mathematics is a very odd
Sort of Person, that reasons about things that have Bulk, Figure, Motion, Number,
Weight, &c. without Arithmetic, Geometry, Mechanics,
Statics, &c.
I must needs say, I have the last Contempt for those Gentlemen, that
pretend to explain how the Earth was framed, and yet can hardly
measure an Acre of Ground upon the Surface of it: And as the
Philosopher
speaks, Qui repente
pedibus illotis ad Philosophos divertunt, non hoc est satis, quod sint
omnino,
αθεώρητοι,
αμγσοι,
αγεωμετρηλοι
sed legem etiam dant, quȃ
Philosophari discant.
The Usefulness of Mathematics in
several other Arts and Sciences is fully as plain. They were looked [p.16] upon by the antient
Philosophers as to the Key to all Knowledge. Therefore Plato wrote upon
his School, Ουδείς
αγεωμέτρητος
εισείτω, Let
none unskilled in Geometry enter; and Xenocrates told
one ignorant in Mathematics,
who desired to be his Scholar, that he was fitter to card Wool, λαζας
γαξ γχ
εχες
Φιλοσοφίας,
You want the
Handle of Philosophy, viz. Geometry. There is
no understanding the Works of the antient Philosophers without it. Theo Smyrnaeus has
wrote a Book, intituled, An Explanation of those things in Mathematics,
that are necessary for the Reading of Plato: Aristotle
illustrates his Precepts, and other Thoughts, by Mathematical Examples;
and that not only in Logic,
&c. but even in Ethics,
where he makes use of Geometrical and Arithmetical Proportion, to
explain commutative and distributive Justice.
Every body knows, that Chronology and Geography
are indispensable Preparations for History; a Relation of Matter of
Fact being a very lifeless insipid thing, without the Circumstances of
Time and Place. Nor is it sufficient for one, that would understand
things thoroughly, that he knows the Topography, that is, the Name of
the Country, where such a Place lies, with those of the near adjacent
Places, and how these lie in respect of one another; but it will become
him likewise to understand the scientifical Principles of the Art: that
is, to have a true Idea of a Place, we ought to know the
Relation
it has to any other Place, as to the Distance and Bearing, its Climate,
Heat, Cold, Length of Days, &c.
which things do much enliven the Reader's Notion of the very Action
itself. Just so, it is necessary to know the technical or doctrinal
Part of Chronology, if a Man would be thoroughly skilled in History, it
being impossible, without it, to unravel the Confusion of Historians. I
remember Mr. Halley
has determined [p.17]
the Day and Hour of Julius
Caesar's Landing in Britain,
from the Circumstances of his Relation. And every body knows, how great
Use our incomparable Historian Mr. Dodwell
has made of the calculated times of the Eclipses, for settling the
Times of great Events, which before were, as to this essential
Circumstance, almost fabulous. Both Chronology and Geography,
and also the Knowledge of the Sun's and Moon's Motions, so far, as they
relate to the Constitution of the Kalendar and year, are necessary to a
Divine; and how sadly some otherwise Eminent have blundered, when they
meddled with things that relate to these, and border on them, is too
apparent.
Nobody, I think, will question the
Interest, that Mathematics have in Painting,
Music, and Architecture,
which are all founded on Numbers, Perspective and the Rules of Light
and Shadows, are owing to Geometry
and Optics:
And, I think, those Two comprehend pretty near the whole Art of Painting, except Decorum and Ordinance;
which are only due Observance of the History and Circumstances of the
Subject you represent: For, by Perspective, may be understood the Art
of designing the Outlines of your Solid, whether that be a Building,
Landskip, or Animal: And the Draught of a Man is really as much the
Perspective of a Man, as the Draught of a Building is of a Building;
tho', for particular Reasons, as because it consists of more crooked
Lines, &c.
it is hard to reduce the Perspective of the former, to the ordinary
established Rules.
If Mathematics
had not reduced Music
to a regular System, by contriving its Scales,
it had been no Art, but enthusiastic Rapture, left to the roving fancy
of every Practitioner. This appears by the extraordinary Pains, which
the Antients have taken to fit Numbers to Three Sorts of Music, the Diatonic, [p.18] Chromatic, and Enharmonic: Which
if we consider with their Nicety in distinguishing their several Modes, we shall be
apt to judge, they had something very fine in their Music, at least,
for moving the Passions with single Instruments and Voices. But Music had been
imperfect still, had not Arithmetic
stepped in once more, and Guido
Aretinus,
by inventing the Temperament, making the Fifth False by a certain
determined Quantity, taught us to tune our Organs, and intermix all the
Three Kinds of the Antients, to which we owe all the regular and noble
Harmony of our modern Music.
As for Civil Architecture
(of Military I shall speak afterwards) there is hardly any Part of Mathematics, but is
some way subservient to it, Geometry
and Arithmetic,
for the due Measure of the several Parts of a Building, the Plans,
Models, Computation of Materials, Time, and Charges; for ordering right
its Arches and Vaults, that they may be both firm and beautiful: Mechanics, for its
Strength and Firmness, transporting and raising Materials: And Optics, for the
Symmetry and Beauty. And I would not have any assume the character of
an Architect without
a competent skill in all of these. You see that Vitruvius requires
these, and many more, for making a complete Architect.
I must own, that should any one set up to practise in any of the
fore-mentioned Arts, furnished only with his Mathematical Rules, he
would produce but very clumsy Pieces. He, that should pretend to draw
by the Geometerical Rules of Perspective, or compose Music
merely by his Skill in harmonical Numbers, would shew but aukward
Performances. In those compos'd Subjects, besides the stiff Rules,
there must be Fancy, Genius, and Habit. Yet, nevertheless, these Arts
owe their Being to Mathematics,
as laying the Foundation of their Theory, and affording them Precepts,
which, being once invented, are securely rely'd [p.19]
upon by Practitioners. Thus many design, that know not a Tittle of the
Reason of the Rules they practice by; and many, no better, perhaps,
than he could have done, that invented the Scale, and the Numbers upon which
their Harmony is founded. As Mathematics
laid the Foundation of these Arts, so they must improve them: And he,
that would invent, must be skilled in Numbers: Besides, it is fit a Man
should know the true Grounds
and
Reasons of what he studies: And he that does so, will certainly
practice in his Art with greater Judgement and Variety, where the
ordinary Rules fail him.
I proceed now to shew the more immediate
Usefulness of Mathematics
in Civil
Affairs. To begin with Arithmetic,
it were an endless Task to relate its several Uses in public and
private Business. The Regulation and quick Dispatch pf both seen
intirely owing to it, The Nations, that want it, are altogether
barbarous, as some Americans,
who can hardly reckon above Twenty. And, I believe, it would go bear to
ruin the Trade of the Nation, were the easy Practice of Arithmetic
abolished: For Example, were the Merchants and Tradesmen obliged to use
of no other than the Roman
way of Notation by Letters, instead of our present. And if we should
feel the Want of our Arithmetic
in the earliest Calculations, how much more in those, that are
something harder? as Interest simple and compound, Annuities, &c. in
which, it is incredible, how much the ordinary Rules and Tables
influence the Dispatch of Business. Arithmetic
is not only the great Instrument of private Commerce, but by it are (or
ought to be) kept the public Accounts of a Nation: I mean those, that
regard the whole State of a Commonwealth, as to the Number,
Fructification of its People, Increase of Stock, Improvement of Lands
and Manufactures, Balance of [p.20]
Trade, Public Revenues, Coinage, Military Power by Sea and Land, &c.
Those that would judge or reason truly about the State of any Nation,
must go that way to work, subjecting all the fore-mentioned Particulars
to Calculation. This is the true Political
Knowledge. In this respect the Affairs of a Commonwealth
differ
from those of a private Family, only in the Greatness and Multitude of
Particulars, that make up the Accounts. Machiavel goes
this way to work in his Account of different Estates. What Sir William Petty, and
several others of our Countrymen, have wrote in Political Arithmetic,
does abundantly shew the Pleasure and usefulness of such Speculations.
It is true, for want of good Information, their Calculations sometimes
proceed upon erroneous Suppositions: But that is not the Fault of the
Art. But what is it the Government could not perform in this way, who
have the Command of all public Records?
Lastly, Numbers are applicable even to
such as depend on Chance;
the Quantity of Probability and Proportion of it in any Two proposed
cases being subject to Calculation as much as anything else. Upon this
depend the Principles of Game. We find Sharpers know enough of this, to
cheat some Men that would take it very ill to be thought Bubbles: And
one Gamester exceeds another, as he had a greater Sagacity and
Readiness in calculating his Probability to win or lose in any proposed
Case. To understand the Theory of Chance
thoroughly, requires a Knowledge of Numbers and a pretty competent one
of Algebra.
The several Uses of Geometry are not
much fewer than those of Arithmetic.
It is necessary for ascertaining of Property both in Planes and Solids,
or in Surveying and Gauging. By it, Land is sold by the Measure, as
well as Cloth: Workmen are paid [p.21]
the due Price of their Labour, according to superficial or solid
Measure of their Work: And the Quantity of Liquors determined for a due
Regulation of their Price and Duty. All which do wonderfully conduce to
the easy Dispatch of Business, and the preventing of Frauds and
Controversies. I need not mention the measuring Distances, laying down
of Plans and Maps of Countries, in which we have daily Experience of
its Usefulness. These are some familiar Instances of things, to which Geometry is
ordinarily applied: Of its Use in Civil,
Military, and Naval
Architecture, we shall speak afterwards.
From Astronomy
we have the regular Disposition of our Time, in a due Succession of
Years, which are kept within their Limits as to the Return of the
Seasons, and the Motion of the Sun. This is no small Advantage for the
due Repetition of the same Work, Labour, and Actions. For many of our
Public, Private, Military, and Country Affairs, Appointments, &c.
depending on the Products of the Ground, and they on the Seasons; it is
necessary that the Returns of them be adjusted pretty near to the
Motion of the Sun: And we should quickly find the Inconveniency of a vague undetermined
Year, if we used that of the Mahumetans,
whose Beginning, and every Month wanders through all the Days of ours
or the Solar Year, which shews the Seasons. Beside, the adjusting of
the Moon's Motion to the Sun's is required for the decent Observation
and Celebration of the Church-Feasts
and Fasts,
according to the antient Custom, and primitive Institution; and,
likewise, for the knowing of the Ebbing and Flowing of the Tides, the
Spring, and Neap Tides, Currents, &c.
So that whatever some People may think of an Almanac where all
these are set down, it is oftentimes the most useful Paper that is
published the same Year with it: Nay, the [p.22]
Nation could better spare all the voluminous Authors in the Term-Catalogue,
than that single Sheet. Besides, without a regular Chronology, there
can be no certain History; which appears by the Confusion amongst
Historians before the right Disposition of the Year, and, at present,
among the Turks,
who have the same Confusion in their History as in their Kalendar.
Therefore, a Matter of such Importance might well deserve the Care of
the Great Emperor,
to whom we owe our present Kalendar; who was himself a great Proficient
in Astronomy.
Pliny has
quoted several things from his Books of the Rising and Setting of the
Stars, Lib.
XVIII. cap.
25, 26, &c.
and Lucan
makes him say,
------- Media inter
praelia semper
Stellarum, coelique plagis, superisque vacavi.
The Mechanics
have produced so many useful Engines, subservient to
Conveniency, that it would be a Talk too great to relate the several
Sorts of them: Some of them keep Life itself from being a Burden. If we
consider such, as are invented for raising Weights, and are employed in
Building, and other great Works, in which no Impediment is too great
for them; or Hydraulic
Engines for raising of Water, serving for great Use and Comfort to
Mankind. where they have no other way to be supply'd readily with that
necessary Element; or such as, by making Wind and Water work for us,
save animal Force, and great Charges, and perform those Actions, which
require a vast Multitude of Hands, and without which every Man's Time
would be too little to prepare his own Aliment, and other Necessaries;
or those Machines, that have been invented by Mankind for Delight and
Curiosity, imitating the Motions of Animals, or other Works of Nature;
we shall have Reason to admire and extol so excellent an Art. What
shall we say [p.23]
of the several Instruments, which are contrived to measure Time? We
should quickly find the Value of them if we were reduced to the
Condition of those barbarous Nations that want them. The Pendulum Clock,
invented and completed by that famous Mathematician
Monsieur Huygens,
is an useful Invention. Is there any thing more wonderful, than several
Planetary Machines,
which have been invented to shew the Motions of the heavenly Bodies,
and their Places at any time? Of which the most ingenious, according to
the exactest Numbers, and true System, was made by the same M. Huygens: To which
we may very justly apply Claudian's
noble Verses upon that of Archimedes:
Jupiter in parvo
cùm cerneret aethera vitro,
Risit, & ad superos talia dicta
dedit:
Huccine mortalis progressa potentia curae?
Jam meus in fragili luditur orbe labr.
Jura poli, rerumque fidem, legesque Deorum
Ecce Syracusius transtulit arte senex.
Inclusus variis famulatur spiritus astris,
Et vivum certis motibus urget opus.
Percurrit proprium menititus Signifer annum,
Et simulata novo Cynthia mense redit.
Jamq; suum volvens audax industria mundum
Gaudet, & humanȃ sidera mente
regit.
Quid falso insontem tonitru Salmonea miror?
AEmula naturae parva reperta manus.
Here I ought to mention the Sciatherical Instruments,
for want of which there was a time, when the Grecians themselves
were forced to measure the Shadow, in order to know the Hour; and, as Pliny (cap. ult. lib. VII.)
tells us, the Romans
made use of an erroneous Sun-dial for Ninety-nine Years, till Q. Marcius Phillipus,
their Censor, set up a better; which, no doubt, at that time, was
thought a [p.24]
Jewel. And at last, that famous Pyramid was set up in the Campus Martius, to
serve for a Gnomon to a Dial marked on the Street. To this Sort of
Engines ought to be referred Spheres,
Globes, Astrolabes, Projections of the Sphere, &c.
These are such useful and necessary things, that alone may recommend
the Art, by which they are made. For, by these, we are able in our
Closet to judge of the Celestial Motions, and to visit the most distant
Places of the Earth, without the Fatigue and Danger of Voyages; to
determine concerning their Distance, Situation, Climate, Nature of the
Seasons, Length of their Days, and their Relation to the celestial
Bodies, as much as if we were Inhabitants. To all these I might add
those Instruments, which the Mathematicians
have invented to execute their own Precepts, for making Observations either
by Sea or Land, Surveying,
Gauging, &c.
The Catoptrics
and
Doptrics
furnish us with Variety of useful Inventions, both for promoting of
Knowledge, and the Conveniencies of Life; whereby Sight, the great
Instrument of our Perception, is so much improved, that neither the
Distance nor the Minuteness of the Object are any more Impediments to
it. The Telescope
is of so vast Use, that, besides the delightful and useful Purposes it
is apply'd to here below, as the descrying Ships, and Men, and Armies,
at a Distance, we have, by its means, discovered new Parts of the
Creation, fresh Instances of the surprising Wisdom of the adorable
Creator. We have, by it, discovered the Satellites of Jupiter, the Satellites and Ring of Saturn, the
Rotation of the Planets about their own Axes; besides other
Appearances, whereby the System of the World is made plain to Sense, as was
before to Reason.
The Telescope
has also improved the Manner of Astronomical
Observations, and made them much more accurate, than it
was possible for them [p.25]
to be before. And these Improvements in Astronomy, have
brought along with them (as ever) correspondent Improvements in Geography. From the
Observation of Jupiter's
Satellites. we have a ready Way to determine the Longitude
of Places on the Earth. On the other hand, the Microscope has not
been less useful in helping us to the Sight of such Objects, as by
their Minuteness escape our naked Eye. By it Men have pursued Nature
into its most retired Recesses; so that now it can hardly any more hide
its greatest Mysteries from us. How much have we learned by the Help of
the Microscope
of the Contrivance and Structure of animal and vegetable Bodies, and
the Composition of Fluids and Solids? But if these Sciences had never
gone further, than by their single Specula
and Lentes
to give those surprising Appearances of Objects, and their Images, and
to produce Heat unimitable by our hottest Furnaces, and to furnish
infallible, easy, cheap, and safe Remedies for the Decay of our Sight
arising commonly from Old Age, and for Purblindness, they had merited
the greatest Esteem, and invited to the closest Study: Especially, if
we consider, that such as naturally are almost blind, and either know
not their nearest Acquaintance at the Distance of a Room's Breadth, or
cannot read, in order to pass their time pleasantly, are, by Glasses
adapted to the Defect of their Eye, set on a Level again with those
that enjoy their Eye-sight best, and that without Danger,m Pain, or
Charge.
Again, Mathematics are
highly serviceable to a Nation in Military
Affairs. I believe this will be readily acknowledged by
every Body. The Affairs of War take in Number, Space, Force, Distance,
Time, &c.
(Things of Mathematical
Consideration) in all its Parts, in Tactics,
Castramentation, Fortifying, Attacking, and Defending. The
Antients had more Occasion for Mechanics in the Art of War than we [p.26] have; Gunpowder
readily producing a Force far exceeding all the Engines, they had
contriv'd for Battery. And this, I reckon, has lost us a good Occasion
of improving Mechanics; the Cunning of Mankind never exerting itself so
much, as in their Arts of destroying one another. But, as Gunpowder has
made Mechanics less serviceable to War; it has made Geometry more
necessary: There being a Force of Resistance in the due Measures and
Proportions of Lines and Angles of a Fortification, which contribute
much towards its Strength. This Art of Fortification has
been much study'd of late, but I dare not affirm, that it has attained
its utmost perfection. And tho', where the Ground is regular, it admits
but of small variety, the Measures being pretty well determined by Geometry and
Experience, yet where the Ground is made up of natural Strengths and Weaknesses, it
affords some Scope for Thinking and Contrivance. But there is another
much harder Piece of Geometry,
which Gunpowder has given us Occasion to improve, and that is the
Doctrine of Projectiles; whereon the Art of Geometry is
founded. here the geometers
have invented a beautiful Theory, and Rules and Instruments, which have
reduced the Casting of Bombs to great Exactness. As for Tactics and Castramentation, Mathematics retain
the same Place in them as ever. And some tolerable Skill in these is
necessary for Officers,
as well as for Engineers.
An Officer,
that understands Fortification, will, caeteris paribus,
much better defend his Port, as knowing, wherein its Strengths
consists, or make use of his Advantage to his Enemy's Ruin, than he
that does not. He knows, when he leads never so small a party, what his
Advantages and Disadvantages in Defending and Attacking are, how to
make the best of his Ground, &c.
And hereby can do truly more Service than another of as much [p.27] Courage, who, for
want of such Knowledge, it may be, throws away himself, and a Number of
brave Fellows under his Command; and it is well, if the Mischief
reaches no further. As for a competent Skill in Numbers, it is so
necessary to Officers,
that no Man can be safely trusted with a Company, that has it not. All
the Business is not to fire Musquets; the managing of Affairs, the
dealing with Agents, &c.
happen more frequently. And the higher the Command is, the more Skill
in all the aforesaid things is required. And I dare appeal to all the
Nations in Europe,
whether, caeteris
paribus, Officers are not advanced in Proportion to their
Skill in Mathematical
Learning; except for sometimes Great Names and Quality carry it;
but still so, as that the Prince depends upon a Man of Mathematical Learning,
that is put as Director to the Quality,
when that Learning is wanting in it.
Lastly, Navigation, which
is made up of Astronomy
and Geometry,
is so noble an Art, and to which Mankind owes so many Advantages, that,
upon this single Account, those excellent Sciences deserve most of all
to be studied, and merit the greatest Encouragement from a Nation, that
owes to it both its Riches and Security. And not only doe the common
Art of Navigation
depend on Mathematics,
but whatever Improvements shall be made in the Architecture Navalis,
or Building of Ships, whether they are designed for Merchant-Ships, or
Ships of War, whether swift running, or bearing a great Sail, or lying
neat the Wind, be desired, these must all be the Improvements of geometry. Ship-Carpenters,
indeed, are very industrious; but in these things they acknowledge
their Inability, confess that their best Production are the Effects of
Chance, and implore the Geometer's
Help. Nor will common Geometry do the
Business; it requires the most abstruse to determine the different
Sections of [p.28]
a Ship, according as it is designed for any of the aforesaid Ends, A French
Mathematician P.
le Hoste has lately endeavoured something in this way: And
tho' it is not free from Errors, as requiring a fuller Knowledge of Geometry; yet is
the Author much to be commended for this, as having bravely designed,
and pav'd the Way for other Mathematicians; and also for the former and
bigger Part of his Book, wherein he brings to a System the Working of
Ships, and the Naval
Tactics, or the regular Disposition of a Fleet in
Attacking, Fighting, and Retreating, according to the different
Circumstances of Wind, Tides, &c.
The great Objection, that is made
against the Necessity of Mathematics,
in the fore-mentioned great affairs of Navigation, the Art Military,
&c. is,
that we see those Affairs are carried on and managed by such, as are
not great Mathematicians; as Seaman, Engineers, Surveyors, Gaugers,
Clock-makers, Glass-grinders, &c.
and that the Mathematicians are commonly speculative retired, studious
Men, that are not for an active Life and Business, but content
themselves to sit in their Studies, and pore over a Scheme, or a Calculation. To
which there is this plain and easy Answer: The Mathematicians have not
only invented and ordered all the Arts above-mentioned, by which those
grand Affairs are managed; but have laid down Precepts, contrived
Instruments and Abridgements so plainly, that common Artificers are
capable of practicing by them, tho' they understand not a Tittle of the
Grounds, on which the Precepts are built. And in this they have
consulted the Good and Necessities of Mankind. Those Affairs demand so
great a Number of People to manage them, that it is impossible to breed
so many good or even tolerable Mathematicians. The only thing then to
be done was to make their Precepts so plain, that they might [p.29] be understood and
practiced by a Multitude of Men. This will best appear by Examples.
Nothing is more ordinary than Dispatch of Business by common Arithmetic, by the Tables of simple and compound
Interest, Annuities, &c. Yet how few Men of
Business understand the Reasons of common Arithmetic, or the
Contrivance of those Tables, now they are made; but securely rely on
them as true. They were the good and the thorough Mathematicians, that
made those Precepts so plain, and calculated those Tables, that
facilitate the Practice so much. Nothing is more universally necessary,
than the measuring of Plains and Solids: And it is impossible top breed
many good Mathematicians, as that there may be one, that understands
all the Geometry
requisite for Surveying, and Measuring of Prisms and Pyramids, and their
Parts, and measuring Frustrums
of Conoids
and Spheroids,
in every Market-Town, where such Work is necessary: The Mathematicians
have therefore incrib'd such Lines on their common Rulers, and Slipping
Rulers, and adapted so plain Precepts for them, that every
Country-Carpenter, and Gauger, can do the Business accurately enough;
though he knows no more of those Instruments, tables and Precepts he
makes use of, than a Hobby-horse. So in Navigation, it
is impossible to breed so many good Mathematicians, as would
be necessary to sail the hundredth Part of the Ships of the Nation. But
the Mathematicians have laid down so plain and distinct Precepts,
calculated necessary Tables, and contrived convenient Instruments, so
that a Sea-man, that knows not the Truths, on which his Precepts and
Tables depend, may practice safely by them. They resolve Triangles
every Day, that know not the Reason of any one of their Operations Seamen
in their [p.30]
Calculations make use of artificial
Numbers, or Logarithms,
that know nothing of their Contrivance: And indeed all those great
Inventions of the most famous Mathematicians had been almost useless
for those common and great Affairs, had not the Practice of them been
easy to those who cannot understand them. From thence it is plain, that
is it those Speculative
Retir'd Men, we owe the Rules, the Instruments, the
Precepts for using them, and the Tables which facilitate the Dispatch
of so many great Affairs, and supply Mankind with so many Conveniences
of Life. They were the Men, that taught the World to apply Arithmetic, Astronomy, and
Geometry,
to Sailing,
without which the Needle would be still useless. Just the same way in
the other parts of Mathematics,
the Precepts that are practiced by Multitudes, without being
understood, were contriv'd by some few great Mathematicians.
Since then, it has been shewn, how much Mathematics improve
the Mind, how subservient they are to other Arts, and how immediately
useful to the Commonwealth, there needs no other Arguments or Motives
to a Government, to encourage them, This is the natural Conclusion from
the Premises. Plato,
in his Republic,
(lib. VII.)
takes care, That,
whoever is to be educated for Magistracy, or any considerable Post in
the Commonwealth, may be instructed first in Arithmetic, then in
Geometry, and thirdly in Astronomy. And however
necessary those Arts were in Plato's
time, they are much more so now: The Arts of War and Trade requiring
much more the Assistance of those Sciences
now, than did then: as being brought to a greater Height and
Perfection. And accordingly we see, these Science are the
particular Care of Princes, that design to raise the [p.31] Force and Power of
their Countries. It is well known, that this is none of the least Arts,
whereby the French King
has brought his Subjects to make that Figure at Sea, which they at this
Time do; I mean, the Care He takes for Educating those appointed for
Sea-service in Mathematical
Learning. For in the Ordonnance
Marine, Title VIII. 'He orders, that there be Professors
to teach Navigation publickly in all the /sea-port Towns, who must know
Designing,
and teach it to their Scholars, in order to lay down the Appearances of
Coasts, &c. They
are to keep their Schools open, and read four times a Week to the
Seamen, where they must have Charts, Globes, Spheres, Compasses,
Quadrants, Astrolabes, and all Books and Instruments necessary to teach
their Art. The Directors of Hospitals are obliged to send thither
yearly two or three of their Boys to be taught, and to furnish them
with Books to be taught, and to furnish them with Books and
Instruments. Those Professors are oblig'd to examine the journals
deposited in the Office of the Admiralty, in the Place of their
Establishment; to correct the Errors in Presence of the Seamen, and to
restore them within a Month.' &c.
King Charles
the Second, who well understood the importance of Establishments of
this Nature, founded one such school in Christ's Hospital, London;
which, I believe, is inferior to none of the French: But 'tis to
be wished there were many more such. His present Majesty, during the
Time of the late War, established a Mathematical Lecture
to breed up Engineers and Officers, as knowing very well the Importance
thereof. And this continued some time after the Peace. And it is
worthy the Consideration of the Wisdom
of the Nation, whether the restoring and continuing this, even in Peace, be note
expedient for the breeding [p.31]
of Engineers who are found so useful and valuable, and so difficult to
be had in Time of War, and so little dangerous in Times of Peace.
Besides the Crowd of Merchants, Seamen, Surveyors,
Engineers, Ship-Carpenters, Artisans, &c. that
are to be instructed in the Practice of such Parts of Mathematics, as are
necessary to their own business respectively, a competent Number of able Mathematicians
ought to be entertained, in order to apply themselves to the Practice;
not only to instruct the former Sort, but likewise to remove those
Obstacles, which such, as do not think beyond their common Rules,
cannot overcome. And no doubt it is no small Impediment to the
Advancement of Arts, that Speculative
Men, and good
Mathematicians, are unacquainted with their particular
Defects, and the several Circumstances in them, that render things practicable of impracticable. But
if there were public Encouragement, we should have skillful
Mathematicians employed in those Arts, who would certainly find out and
remedy the Imperfections of them. The present Lords Commissioners of
the Admiralty, knowing that there are still two great Desiderata in
Navigation, to wit, The
Theory of the Variation of the magnetical Needle and a Method of finding out the
Longitude of any Place, that may be practicable at Sea by
Seamen, and being sensible, of what Importance it would be to find out
either of them, have employed a very fit Person, the ingenious Mr. Halley, who has
joined an intire Acquaintance in the Practice, to a full and thorough
knowledge of the more abstruse Parts of Mathematics. And
now that he is returned, it is not doubted, but he will satisfy those
that sent him, and in due time, the World too, with his Discoveries in
both those Particulars, and in [p.33]
many other that he has had Occasion to make. And where a long Series of
Observations and Experiments is necessary, he has, no doubt, laid such
a Foundation, as that After-Observers may gradually perfect them. If it
were not for the Coasts where he touched, and by them others whose
Relation to the former is known, the Nation is more than triply paid:
And those who sent him, have, by this Mission, secured to themselves
more true Honour, and lasting Fame, than by Actions, that, at first
View, appear more magnificent.
The next thing that is necessary for the
Improvement of Mathematical
Learning, is, That Mathematics be more generally studied
at our Universities
than hitherto they have been. From those Seminaries the State justly
expects and demands Those who are acquainted both with the Speculation and Practice. In those
are all the Encouragements to them imaginable, Leisure and Assistance.
There are still at hand Books and Instruments; as also other Scholars
that have made equal Progress, and many be Comrades in Study; and the
Direction of the Professors. There are also in Perfection all the
Incitements to this Study; and especially an Acquaintance with the
Works of the Antients, where this
Learning is much recommended. There other Faculties are
studied, to which it is subservient. There also are the Nobility and
Gentry bred; who, in due time, must be called to their Share in the
Government of the Fleets,
Army, Treasury, and other public Employments, where Mathematical Learning
is absolutely necessary, and, without which, they, tho' of great
natural Parts, must be at the Mercy and Discretion of their Servants
and Deputies; [p.34]
who will first cheat them, and then laugh at them. And not
only public Employments, but their private Concerns, demand
Mathematical Knowledge. If their Fortunes lie in Woods, Coal, Salt,
Manufactures, &c.
the Necessity of this Knowledge is open and known: And, even in
Land-Estates, no Undertaking for Improvement can be securely relied
upon without it. It not only makes a Man of Quality and Estate his
whole Life more illustrious, and more useful for all Affairs,
(as Hippocrates
says
&c.) but in particular, it is best Companion for Country Life.
Were this once become a fashionable Study, (and the 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
rendering them the more serious, diligent, curious; taking them off
from the more fruitless and airy Exercises of the Fancy, which they are
apt to run into.
The only Objection I can think of, that
is brought against these Studies, is, That Mathematics require a
particular Turn of Head, and a happy Genius, that few People are
Masters
of; without which all the Pains bestowed upon the Study of them are in
vain: They imagine, that a Man
must be born a Mathematician. I answer, That this Exception is common
to Mathematics and other Arts. That there are Persons that have a
particular Capacity and Fitness to one more than another, every body
owns: And, from Experience, I dare say, it is not in any higher Degree
true concerning Mathematics, than the others. A Man of good Sense and
Application is the Person that [p.35]
is by Nature fitted for them; especially if he begins betimes: And, if
his Circumstances have been such, that this did not happen, by prudent
Direction the Defect may be supplied, as much as in any Art whatsoever.
The only Advantage this Objection has, is, That it is on the Side of
Softness and Idleness, those powerful Allies!
There is nothing further remains, Sir,
but that I give you my Thoughts in general concerning the Order and Method of studying Mathematics; which
I shall do very briefly, as knowing that you are already acquainted
with the best Methods; and others with you may have them easily from
the best and ablest Hands.
First, then, I lay down for a Principle,
That nobody at an University
is to be taught the Practice of any Rule without the true and solid
Reason and Demonstration of the same. Rules without Demonstration must
and ought to be taught to Seamen,
Artisans, &c. as I have already said; and Schools
for such People are fit in Sea-Ports and Trading-Towns; but it is far
below the Dignity of an University,
which is design'd for solid and true Learning, to do this. It is from
the Universities that they must come, who are able to remedy the
Defects of the Arts; and therefore nothing must come, who are able to
remedy the Defects of the Arts; and therefore nothing must be taken on
Trust there. Seamen
and Surveyors,
&c. remember their Rules, because they are perpetually
practicing them; but Scholars,
who are not thus employ'd, if they know not the Demonstration of them,
presently forget them.
Secondly, No Part of Mathematics ought
to be taught by Compendiums.
This follows from the former. Compendiums
are fit to give a general and superficial
Knowledge, not a thorough one. It is Time, and not the Bulk of Books,
we ought to be sparing of: And I appeal to any Person of [p.36] Experience, whether
solid Knowledge is not acquir'd in shorter time by Books treating fully
of their Subjects, than by Compediums and Abridgements,
From hence it follows, that the Elements of Arithmetic and Geometry are to be
taught. Euclid,
in his Thirteen Books of Elements, gives us both: but our present Way
of Notation supersedes some of those of Arithmetic, as
demonstrating the Rules from the Operations themselves. There remain
then the first Six Books for the Geometry
of Planes,
and the last Three for Stereometry.
The rest ought to be read in their own Place, for the Perfection of Arithmetic. In
teaching these, Care ought to be taken to make use of such Examples, as
suit with the Condition of the Scholar: For Instance, Merchants Accompts
and Affairs
for Examples of the Operations of Arithmetic,
to one that is afterwards to have a Concern that way; whereas, to a Man
of the first Quality, Examples from the Increase and Decrease of the
People, or from Land
or Sea Force,
and from the Tactics,
ought to be proposed. For, it is certain, nothing makes one tir'd
sooner, than the frivolous and trifling Examples, that are commonly
brought for the Exercise of the Rules of Arithmetic and Geometry; tho' this
is common to them with the other Arts, as Grammar, Logic,
&c.
The Manner of Writing of the
Mathematicians of This and the former Age makes Trigonometry, with
the Manner of Constructing Tables, &c.
almost Elementary:
And the Practical
Geometry, commonly so call'd, is very fit to come next,
as an elegant Application of the Elements
of Geometry
to Business, as Surveying,
Gauging, &c.
After the Elements of Spherics which are
perfectly well-handled by Theodosius,
a full Insight into the Principles of astronomy will be
necessary.
[p.37]
Mechanics
come next to be read, which are the Ground of a great Part of natural
Learning; and, afterwards, Optics,
Catoptrics, and
Dioptrics.
But none of these, except the Elements,
can be fully understood, until one is pretty well skill'd in Conic Sections: And
all these are made more easy by some tolerable Skill in Algebra, and its
Application to Geometry.
These Foundations being laid, and one
may, with great Ease, pursue the Study of the Mathematics, as his
Occasions require; either in its abstract Parts, and the more recondite Geometry,
and its Application to Natural Knowledge; or in Mechanics; by
prosecuting the Statics,
Hydrostatics, Ballistics, &c.: Or in Astronomy, by its
Application to Geography,
Navigation, Gnomonics, Astrolabes, &c. But, in
most of these, a particular Order is not necessary: Any one may take
That first, which he is most inclined to.
I shall not offer you any Advice
concerning the Choice of Books; but refer you (if you want any) to the
Direction of those who are eminent among you in this Part of Learning.
I ask your Pardon for the Omission of Ceremony in these
Papers; having followed rather the ordinary Way of Essay, than Letter. And,
wishing you good Success with your Studies, I am,
S
I R,
Your Friend
and Servant.
25 Novemb.
1700