George Atherton Aitken.

The life and works of John Arbuthnot, M.D., fellow of the Royal College of Physicians online

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of fluids and solids? But if these sciences had never gone


further, than by their single sjiecula and lenses to give those
suri^rising appearances of objects and their images, and to
produce heat ununitable 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 : esjDecially 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 eyes set on a level again with those that enjoy their
eyesight best, and that without danger, pain or charge.

Again, mathematics are highly serviceable to a nation in
military affairs. I believe this will be readily acknowledged by
everybody. The affairs of war take in number, space, force,
distance, time, &c. (things of mathematical consideration) in all
its parts, in tactics, castrametation, fortifying, attacking, and de-
fending. The ancients had more occasion for mechanics in the
ai"t of war than we have ; gunpowder readily producing a force
far exceeding all the engines they had contrived for battery.
And this I reckon has lost us a good occasion of improving our
mechanics : the cunning of mankind never exerting itself so
much as in their aits of destroying one another. But, as
gunpowder has made mechanics less serviceable to war, it has
made geometiy 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 toward its strength.
This art of fortification has been much studied of late, but I
dare not affirm that it has attained its utmost perfection. And
though 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
geometiy, which gunpowder has given us occasion to improve,
and that is the doctrine of projectiles ; whereon the art of
gunneiy is founded. Here the geometers have invented a
beautiful theoiy, and rules and instruments, which have re-
duced the casting of bombs to great exactness. As for tactics
and castrametation, 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 cacleris paribus much better defend his post,
as knowing wherein its stiength 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 ad-
vantages and disadvantages in defending and attacking are, how
to make the best of his ground, &c., and hereby can do truly
moie sei-vice 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 is well if the
mischief reaches no further. As for a competent skill in
nvunbers, 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 muskets ; 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 cactcrts paribus officers are not advanced in proportion
to their skill in mathematical learning ; except that 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 does the common art of navigation
depend on mathematics, but whatever improvements shall be
made in the Architectura Navalis or building of 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 geometiy. Ship-carpenters indeed are veiy industrious ; but
in these things they acknowledge their inability, confess that
their best productions 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 a ship, according as it is designed for any of the aforesaid
ends. A French mathematician, P. Le Hoste, has lately en-
deavoured something in this way ; and though 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 hravely
designed, and paved the way for other mathematicians ; and
also for the former and bigger part of liis 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 forementioned great affaii's of navigation,
the art military, &c. is, that we see those affairs are carried on
and managed by such as ai*e not great mathematicians ; as
seamen, engineers, surveyors, &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 affah's are managed ; but have laid down
precepts, contrived instruments and abridgments so plainly, that
common artificers are capable of practising by them, though
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 peoj)le to manage them, that it is impossible to breed
so many good or even tolerable mathematicians. The only
thmg then to be done was to make then 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 arithmetic, by the tables
of simple and compound interest, annuities, &c. Yet how few
men of business understand the reasons of common aiithmetic
or the contrivance of those tables, now they are made ; but
securel}^ rely on them as true. They were the good and the
thorough mathematicians, that made those precepts so plain and
calculated those tables that facihtate the practice so nnich.
Nothing is more universally necessary than the measuring of
plains and solids ; and it is unpossible to breed so many good
mathematicians, as that there may be one that understands all
the geometiy requisite for surveying and measuring of prisms
and pyramids, and their parts, and measuring frustums of


conoids and spheroids, in eveiy market-town where such work
is necessaiy : the mathematicians have therefore inscribed such
lines on their common rulers, and slipping rulers, and adapted
so plain i)rocepts to them, that everj^ countiy carpenter and
ganger can do the business accurately enough ; though he
knows no more of those insti-uments, 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
necessaiy to sail the hundredth paii 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 seaman, that knows not the truths on
which his precepts and tables depend, may practise safely by
them. They resolve triangles eveiy day, that know not the
reason of any one of their operations. Seamen in theu" calcula-
tions make use of artificial numbers or logarithms, that know
nothing of their contrivance : and indeed all those great inven-
tions 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.
Froni hence it is plain, that it is to those speculative retired
men we owe the rules, the insti-uments, 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 sailing, without which the needle
would be still useless. Just the same way in the other parts of
mathematics, the precepts that are practised by multitudes,
without being understood, were contrived by some few great

Since then it has been shewn, how much mathematics
improve the mind, how subservdent 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 these premises. Plato in
his BcpuNic (Lib. VII) takes care that whoever is to be
educated for magistracy, or any considerable post in the com-
monwealth, may be instructed first in arithmetic, then in
geometry, and thirdly in astronomy. And however necessary
those ai-ts were in Plato's time, they are much more so now ;
the aiis of war and trade requiring much more the assistance of


those sciences now than they did then, as l)eing brouglit to a
greater height and perfection. And accordingly we see these
sciences ai-e 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 nifithematical learning. For in the
Ordonnance Marine, tit. xaii., he orders that there 'be pro-
fessors to teach navigation publicly 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, com-
passes, quadrants, astrolabes, and all books and instruments
necessaiy to teach then- 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 and instruments.
Those professors are obliged to examine the journals deposited
in the Office of 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 it is 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, 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 comj^lete number of able mathe-
maticians ought to be entertained, in order to apply themselves
to the practice ; not only to mstruct 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 i-ender things practicable or impracticable. But if
there were public encouragement, we should have skilful mathe-
maticians employed in those ai-ts, 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 theoiy of the
variation of the magnetical needle, and a method of studying
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 veiy fit person, the
ingenious Mr. Halley, who has joined an entire 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 w^orld too, with his discoveries in both those
pai-ticulars, and in many other, that he has had occasion to
make. And where a long series of observations and experi-
ments is necessary, he has no doubt laid such a foundation, as
that after-obsei-vers 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 than triply paid ; and
those who sent him have by his 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

' Edmund Halley (1656 1742) a General Chart, showing at one view
was appointed by William the the variation of the Compass in all those
Third Commander of the ' Para- seas where the English narigators ivere
mour Pink,' in 1698, with orders acquainted. Soon aftcrwai-ds Halley
to seek the discovery of the rule of was sent on another mission. The
the variation of the needle. Owing question of finding the longitude
to troubles with the crew, he re- at sea was a problem which con-
turned in 1699, but soon set out stantly interested him. Afterwards
again, in charge of two ships. he became Savilian Professor of
After a long cruise, he again Geometry at Oxford, Secretary to
reached England in September the Royal Society, and Astronomer
1700; and next year he published Koyal.


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 imaginaljle, leisure
and assistance. There are still at hand books and instruments,
as also other scholars that have made equal progress, and may
l^e comrades in study, and the direction of the professors.
There are also in perfection all the incitements to this study,
and especially an acquaintance wdth the works of the ancients,
where this learning is so 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, treasuiy,
and other public employments, where mathematical learning is
absolutely necessaiy, and without which they, though of never
so great natural parts, must be at the mercy and discretion of
their sei-vants and deputies ; who will first cheat them, and
then laugh at them. And not only public employments, but
their private concerns demand mathematical knowledge. If
then- 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\ 'laropirji be /ieXeVw o-ol w Trm,

FfcofifTpiKTjs Kcu 'Api^jUJjfTtoj. ov yap fxovov (Tfo KCLi Tov ^Lov fVKkta Koi e-ni
TtoWa xpW'-P-"^ ^^ uv6pcoiTLi'r]v poiprjv eniTeXecrei, uWa Koi ttju ^vxr/v

n^vreprjv re Kai Tr]\avy€(TT€pr]v, &c.) but in particular, it is the best
companion for a country life. Were this once become a fashion-
able 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 the 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 tm-n 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 a mathematician.

» In this and other quotations I given by Arbnthnot, correcting only
liavc, as a rule, retained the reading obvious mistakes.


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, everybody
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 apphcation is the person that is by nature
fitted for them ; especially, if he begins betimes ; and if his
circvmistances 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 two
powerful allies.

There is nothing further remains, Sii*, but that I give you
my thoughts in general concerning the order and method of
studying mathematics ; which I shall do veiy 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 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 designed 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.
Seamen and surveyors, &c., remember their rules, because they
are perpetually practising them ; but scholars, who are not thus
employed, 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 experi-
ence, whether solid knowledge is not acqviired in shorter time
by books treating fully of then* subjects, than by compendiums
and abridgments.

From hence it follows, that the elements of arithmetic and




geometry are to be tatight. 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 plains, 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' accounts
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 tired sooner, than the frivolous and trifling examples that
are commonly brought for the exercise of the rules of arith-
metic and geometry ; though 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 its tables,, almost elementaiy ; and the prac-
tical geometry, commonly so called, 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.

Mechanics come next to be read, which are the ground of a
great part of natural learning : and afterwards optics, catop-
trics, and dioptrics.

But none of these except the elements can be fully under-
stood until one is pretty well skilled in conic sections ; and
all these are made more easy by some tolerable skill in algebra,
and its application geometry.

These foundations being laid, anyone 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, gno-
raonics, astrolabes, &c. But in most of these a particular


order is not necessary. Anyone may take that first which he
is most inclined to.

I shall not offer you any advice concerning the choice of
))ooks, 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,

Your Friend and Servant.

25 November, 1700.

F f 2


What am I? how produced? and for what end?

Whence drew I bemg? to what period tend?

Am I the abandoned orphan of blind chance,

Dropt by wild atoms in disordered dance?

Or from an endless chain of causes wrought?

And of unthinking substance, born with thought?

By motion which began without a cause,

Supremely wise, without design or laws.

Am I but what I seem, mere flesh and blood ;

A branching channel, with a mazy flood ?

The purple stream that through my vessels glides,

Dull and unconscious flows like common tides :

The pipes through which the circling juices stray,

Ai'e not that thinking I, no more than they :

This frame, compacted, with transcendent skill.

Of moving joints obedient to my will ;

Nursed from the fruitful glebe, like yonder tree.

Waxes and wastes ; I call it mine, not me :

New matter still the mouldering mass sustains.

The mansion changed, the tenant still remains :

And from the fleeting stream repaired by food,

Distinct, as is the swimmer from the flood.

What am I then ? sure, of a nobler birth,

Thy jDarents right, I own a mother, earth ;

But claim superior lineage by my Sire,

Who warmed the unthinking clod with heavenly fire :

' 'A Poem, London 1734. Adver- contains some thoughts of Monsieur

Online LibraryGeorge Atherton AitkenThe life and works of John Arbuthnot, M.D., fellow of the Royal College of Physicians → online text (page 39 of 47)