George Atherton Aitken.

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

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mined the height of the atmosphere, as far as it has any
sensible density ; which agrees exactly with another obsei^vation
of the duration of the twilight. Air and water make up the
object of the hydrostatics, though denominated only from the
latter, of which the principles were long since settled and
demonstrated by Archimedes, in his book -ntpi. rav oxovixtvco",
where are demonstrated the causes of several suq^rising pheno-
mena of nature, depending only on the equilibrium of fluids,
the relative gravities of these fluids, and of solids swimming or
sinkmg 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 geometiy.

Next, if we descend to the animal kingdom, there we see the
brightest strokes of divine mechanics. And whether we consider
first the animal economy 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 exter-motions, and
the instiniments 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 Anhnalium,
Steno in his admirable Myologlac specimen, and other mathe-
matical 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 geometiy is in such speculations. The only organ of
an anhixal body whose structure and manner of operation is

in 1672, but its novelty provoked optical lectures which he had pre-
so much opposition that Newton i^arod for the press,
put on one side for a time the



USEFULNESS OF MATHEMATICAL LEARNING. 417

fully understood, has been the only one which the geometers
have taken to their share to consider. It is 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 geometiy found it out, and put it past dispute, though
they api)lied themselves particularly to this, and valued them-
selves on it ; and Galen pretended a particular divine com-
mission to treat of it. Nay, notwithstanding the full discovery
of it, some go on in copying then- jaredecessors, 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 viyes of the muscles) is as subject to strict mathematical
disquisition as anything whatsoever ; and even in the theoiy
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 mathematics are necessary for the
study of medicine itself, for which I could brmg quotations out
of their works. Among the letters that are ascribed to Hippo-
crates, there is one to his son Thessalus, recommending to hun
the study of arithmetic and geometry, as necessaiy to medicine.
Galen, in his book intitled, otl aptaros larpbs Ka\ (pi\6a-o(pos,

begins, Olov n. 7ren6v6aaiv ol noKXol rav a6\r]Twv, (TnOvfiuiivTis fiiv
^OXvuniovlKcn yeveaSat, fiT]8iv Se nparTeiv, uy tovtov Tv^fi", (niTrjdevovTei,
TOiovToy Tt Kcu Tols TToXXo'is Tcov luTpoiv <Tvpl^e^t]Kev' (TTaivovcTi pel/ yap
'l7r7roKpdTT]v Ka\ irparov inravTuiv ijyovvraL' yeveadcu 8e avroi/s iv opolois
eK€iP(0 ndvTa pakXov rj tovto npaTTova-i, ol pev yap ov piKpav polpav els
larpiKijv (^aal ava^nXXfadat rqv u<TTpovoplav, Kai 8r]Xov6ri ti)v TavTrjs
rjyovpevTjp e^ 6.viiyKr]s TeayptTpiav. ol 8e ov poPov civto). perepx^ovrai rotrav
ovberepov, ciXXa Km toIs periovai. pepcpoinai. If one of the reaSOns of

the ancients for this be now somewhat unfashionable, to wit,
liecause 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 w^ant 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. Secondly, if we consider, not
only the animal economy in general, but likewise the wonderful
structure of the different sorts of animals, according to the

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41 8 WORKS OF DR. ARBUTHNOT.

different puiposes for which they were designed, the vai'ious
elements they inhabit, the several ways of jirocuring 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 a pity that the qualities of an
expert anatomist and skilful geometer have seldom met in the
same person. When such a one shall appear, there is a whole
Urra incognita of delightful knowledge to employ his time, and
reward his industiy.

As for the other two kingdoms, Borelli and other mathe-
matical men seem to have talked veiy clearly of vegetation ;
and Steno, another mathematician, in his excellent treatise Df.
Solido intra Solidum naturaliter contento, has ajjplied this part of
learning veiy 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 entirely owing to
the geometers ; who have demonstrated its laws both in hard
and elastic bodies ; shewed how to measure its qviantity, 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 imiform, 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
cui-ve of projectiles ; the descent and weight of heavy bodies
when they lie on inclined plains ; the theoiy of the motion of
pendulous bodies, &c.

From what I have said, I shall draw but one corollaiy, that
a natural philosopher without mathematics is a very odd sort of
a person, that reasons about things that have bulk, figure,
motion, number, weight, &c., without arithmetic, geometiy,
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 PMlosophos diucrtunt, non hoc est satis,
Quad sint omnino d6ecoi)i]To-, tiniwaoi, dyeuLiiJ.eTfjt]Toi, scd legem etiani
dant, qua Fhilosophari diseant.

The usefulness of mathematics in several other arts and
sciences is fully as plam. They were looked upon by the



USEFULNESS OF MATHEMATICAL LEARN h\G. 419

ancient philosophei-s as the key to all knowledge. Therefore
Plato wrote upon his school, Oi'if li ayewfieVp^ros (liTLTcoj ' let none
unskilled in goometiy enter ' ; and Xenocrates told one ignorant
in mathematics, who desired to be his scholar, that he was fitter
to card wool, Xo/?<W -yfi,; oiV e^*'^ <I>iXo(TO(J!)ia?, ' you want the
handle of philosophy,' viz. geometry. There is no understanding
the works of the ancient philosophers without it. Theo Smyr-
naeus has wrote a book entitled, An explanation of those things
in 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 arith-
metical propoi-tion, to explain commutative and distributive
justice.

Eveiybody knows that chronology and geography are in-
dispensable preparations for history : a relation of matter of
fact being a veiy Kfeless 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 ari : that is, to have a true idea
of a place, we ought to know tlie relation it has to any other
place, as to the distance and bearing, its climate, heat, cold,
leng-th 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 chronologj'^, if a man
would be thoroughly skilled in histoiy, it being impossible
without it to uni-avel the confusion of historians. I remember
Mr. Halley has determined the day and hour of Julius Caesar's
landing in Britain, from the circumstances of his relation. And
eveiybody knows how great use our incompai^able 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 faljulous. 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 calendar
and year, are necessary to a divine, and how sadly some other-

^ Henry Dodwell published at obiterquedeCyclo Judaeoruinaetate
Oxford, in 1701, his ' De veteribus Christi, dissertationes decern.' The
Graecorum Romanorumque Cyclis, second part is dated 1 702.

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420 WORKS OF DR. ARBUTHNOT.

wise 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 ojDtics ; 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 histoiy 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, landscape, 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
Jiuilding ; though 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 ancients have
taken to fit numbers to three sorts of music, the diatonic,
chromatic, and enhaimonic : which if we consider with their
nicety in distinguishing their several modes, we shall be apt to
judge they had something veiy 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 Guide 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 music.

As for civil architecture (of military I shall speak afterwards)
there is hardly any part of mathematics but is some way
subsei'vient 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 com-
petent skill in all of these. You see that Vitruvius requires



USEFULNESS OF MATHEMATICAL LEARNING. 421

these and many more for making a complete architect. I must
own, that should anyone set up to practise in any of the
fore-mentioned arts, furnished only with his mathematical rules,
he would produce but veiy clumsy i^ieces. He that should
pretend to draw by the geometrical rules of perspective, or
compose music merely by his skill in harmonical numbers,
would shew })ut awkward performances. In those composed
subjects, besides the stiff rules, there must be fancy, genius,
and halnt. 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
lelied upon by practitioners. Thus many design, that know
not a tittle of the reason of the rules they practise by ; and
many no better qualified in their way compose music, bettei-
perhaps than he could have done that invented the scale, and
the numbers upon which their haiTnony is founded. As mathe-
matics 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
practise in his art with greater judgment 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 pubhc and private
business. The regulation and quick dispatch of both seem
entirely owing to it. The nations that want it are altogether
barbarous, as some Amencans, who can hardly reckon above
twenty. And I believe it would go near to ruin the trade of
the nation, were the easy practice of anthmetic abolished ; for
exami^le, were the merchants and tradesmen obliged 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
arithmetic in the easiest 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 oi'dinary
rules and tables influence the disjiatch 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,



432 WORKS OF DR. ARBUTHNOT.

improvement of lands and manufactures, balance of 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 poHtical
knowledge. In this respect the affairs of a commonwealth
<liffer from those of a private family, only in the greatness and
multitude of particulars that make uj? 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 infoi'mation, 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 ax'e applicable even to such things as seem
to be governed 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 anything
else. Upon this depend the princij)les 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 under-
stand the theory of chance thoroughly, requires a great knowledge
of numbers, and a pretty competent one of Algebra.

The several uses of geometiy are not much fewer than those
of arithmetic. It is necessary for ascerfaining of property both
in plains and soKds, or in surveying and gauging. By it land
is sold by the measure as well as cloth ; workmen are paid 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 geometiy
is ordinarily applied ; of its use in civil, military, and naval
architecture we shall speak afterwards.



USEFULNESS OF MATHEMATICAL LEARNING. 423

From astronomy we have the regular disposition of our time,

in a due succession of years, which are kept within their hmits

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 pul^lic, private,

military, and the countiy 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 incon-

veniency of a vague undetermined year, if we used that of the

Mahometans, 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 obsei-vation and celebration of the

church-feasts and fasts according to the ancient custom and

primitive institution ; and Hkewise for the kno%ving of the

eb])ing and flowing of the tides, the spring and neaj) tides,

cuiTents, &c. So that whatever some people may think of an

almanac where all these are set down, it is sometimes 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, w^ho

have the same confusion in their histoiy as in their calendar.

Therefore a matter of such impoiiance might well desen^e the

care of the great Emperor, to whom we owe our present

calendar ; 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, supeiisque vacavi.

The mechanics have produced so many useful engines, sub-
sei-vient to conveniency, that it w^ould be a task 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



424 WORKS OF DR. ARBUTHNOT.

comfort to mankind, where they have no other way to be
supiiHed readily with that necessaiy 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 eveiy 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, miitating the motions of animals, or other
works of nature ; we shall have reason to admu-e and extol so
excellent an art. What shall we say 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 in-
vented and completed by that famous mathematician Monsieur
Huygens is an useful invention. Is there anything more wonder-
ful than several planetaiy 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 cum cerneret aethera vitro,

Risit, et ad superos talia dicta dedit :
Huccine mortalis progressa potentia curae ?

Jam meus in fragili luditur orbe labor.
Jura poli, rerumque fidem, legesque deorum

Ecce Syracusius transtulit arte senex.
Inclusus variis famulatur spiritus astris,

Et vivum certis motibus urget opus.
Pereurrit proprium mentitus signifer annum,

Et simulata novo Cynthia mense redit.
Jamque suuni volvens audax industria mundum

Gaudet, et humana sidera mente regit.
Quid falso insontem tonitru Salnionea miror?

Ae nulla naturae parva reperta nianus.

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 PHny (cap. ult. Lib. VII.) tells us, the Eomans made
use of an erroneous sun-dial for ninety-nine years, till Q. Martins
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 Camjms Martins, to seive for a gnomon to a



USEFULNESS OF MATHEMATICAL LEARNING. 425

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 necessaiy 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, suiveying,
gauging, &c.

The catoptrics and dioptrics 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 imj^roved, 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 applied to here below, as the desciying
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 it was before to reason. The telescope has also
improved the manner of astronomical obsei-vations, 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 obsei*vation of Jupiter's satellites, we
have a ready way to determine the longitude of places on the
earth. On the other hand, the microscojie has not been less
useful in helping vis 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 mysteiies 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



Online LibraryGeorge Atherton AitkenThe life and works of John Arbuthnot, M.D., fellow of the Royal College of Physicians → online text (page 38 of 47)