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physician, and at the Royal Society ; which occasioned an interruption in his astronomical ob-
servations. In this situation he died Nov. 6, 1771, at 76 years of age j his, death having been
occasioned by a fall he received a short time before, in going rather too hastily from his in-
strument to the clock, in observing the sun's meridian altitude.

In his disposition, Dr. Bevis was lively, amiable and liberal; extending his services to all de-
serving objects, under any kind of embarrasments.

♦ Alexis Claud Clairaut, F.R.S. and member of the French Academy of Sciences, &c. was a
most respectable mathematician. He was bom May 13, 1713, at Paris, where his father was a
teacher of mathematics. He was it seems a kind of premature genius, which seconded by his



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. Jig

the Royal Academy of Sciences at Paris. "S" 445, p. 1 9. Translated from
the Latin.

According to Newton's Princip. (cor. 3, prob. 91, lib. 1, and prop. 19, lib. 3)
if an elliptic spheroid, consisting of fluid and homogeneous particles, mutually

father's great attentions, produced very early and extraordinary effects. Having learned the letters
of the alphabet from the diagrams in Euclid's Elements, he could read at 4 years of age, and even
write tolerably well. In a similar degree of advance he passed through the mathematical sciences of
arithmetic, algebra, geometry, &c. so as to master Guisnee's application of algebra to geometry, at
9 years of age. At 10 he studied I'Hopital's Conic Sections, and soon after the Analyse des Infini-
ments Petits, of the same author. At 12 he astonished the Academy of Sciences, by reading to them
his discovery of four curves of the third order, by means of which may be found any number of
mean proportionals between two given lines. And at 13 he laid the foundation of his excellent work
on Cunes of a Double Curvature, printed 3 years after.

The same year, 1726, our young author formed a juvenile society, by associating together a number
of ingenious youths like himself, at once for improving themselves and the mathematical sciences;
among whom were several who afterwards became some of the most respectable members of the
Royal Academy of Sciences; of which academy young Clairaut himself was admitted a member at
18 years of age, being 3 years below the limit prescribed by a regulation of the academy, a regula-
tion which tliey dispensed with in this instance on account of his surprising merit. The same year he
presented to the academy two ingenious memoirs of his own inventions. Soon after this, he accom-
panied M. Maupertuis to Basle on his visit to John Bernoulli ; and on his return, he found the academy
much occupied about the question concerning the figure of the earth; in consequence not long after
he and Maupertuis, retiring to Mount Valerien, formed the project of the measurements at the
polar circle, in which both of them bore so conspicuous a part. In this retreat it was, that the Mar-
chioness of Chatelet, having resolved to learn the science of geometr)- from Clairaut, attended him
tliere to receive her lessons; which gave occasion to his composing his pleasant little treatise on geo-
metry. On the question too of the figure of the earth, about this time, he wrote several interesting
memoirs. The delicate observations of Mairan on the lengths of pendulums, gave occasion to Clairaut
to present a memoir on their oscillations. And the discovery of Bradley on the aberration of the
fixed stars, gave also occasion to Clairaut's presenting a valuable calculation on that subject, in which
he made improvements, by extending his views to that of the planets also, dependent on the same
cause.

Several other memoirs, on various subjects, as, the annual parallax of the stars, the nature of the
refraction of light, conduct us to his still more important labours, in the application of the geome-
trical calculus to the profoundest considerations in physics and astronomy. This produced his work
on the theory of the figure of the earth on hydrostatical principles; in which he considered all the
circumstances and states of the earth, as to fluidity and rigidity. An-j next his theory of the moon,
in which he at length detected a subtile error, which had been committed bv all the best calculators
on that delicate subject. After a long continued labour on this object, in 1751 he carried the prize
proposed on the subject by the Academy of Petersburg. Also, in 1754 came out the first edition of
his Lunar Tables; and in 1765 was given a second edition of the same corrected; to which was added
the piece containing the theory which had gained the Petersburg prize.

During those labours, Clairaut composed his elements of algebra, which appeared in 1746; these
elements are in the same easy and familiar stile as those of his geometry, beforementioned. In 175*



120 PHILOSOPHICAL TRANSACTIONS, [aNNO 1737.

attracting each other in the inverse ratio of the squares of the distances, be
revolved about the axis xa, fig. 12, pi. 4, by which the columns ce, on, ca, in
that spheroid, may be in equilibrio, and so the spheroid may always have the
same figure, the gravity at any point of the surface n must necessarily be in-
versely as the radius cn.

To know therefore whether the spheroid has this property, let us inquire what
attraction any corpuscle n of the whole spheroid suffers, according to the direc-
tion CN ; and fnjm that attraction let us deduct that part of the centrifugal force
which proceeds from the rotation of the spheroid acting in the direction cn;
and then see whether the remaining force is proportional to — . We shall first

then investigate the following problem : and as we intend to apply our disco-
veries to the spheroid of the earth, which all agree to be very little different
from a sphere, our computations must be adapted to those spheroids which
have the smallest difference between the two axes.

Prob. I. To find the attraction, which the spheroid AEae, differing very
little from a sphere, exercises on a corpuscle at the pole a.

For the solution of this problem we may repeat cor. 2, prop. Ql, of the
Principia, by which we learn the manner of finding the attraction of any sphe-
roid, viz. by substituting in the general value of ce, a quantity differing infi-
nitely little from AC; and as in that case the problem becomes much easier,
we may solve it in the following manner.

Let AMDad be a sphere, to the radius AC ; we must find the attraction of the
space arising from the rotation of adce, which attraction added to the attrac-
tion of the sphere, will give the attraction sought.

To find the attraction of the space arising from the rotation of anegdm, call
AC, r; DE, ar; ap, u; then, from the nature of the ellipse, nm := a\/2ru — uu,
and from the nature of the circle am = ^/2ru. But the space arising from

he produced his work on the Determination of the Terrestrial Orbit, in which are considered the
perturbations caused by the action of the other planets. And some time after he successfully applied
the same principles to the theory of the celebrated comet of 1759; the result of which calculations
are given in a work which appeared in 176O. And when, in 1758, the academy lost M. Bouguer,
a pension of 3000 livres, which he enjoyed from the Marine Board, was divided between Monnier
and Clairaut, which new engagement produced from his pen an excellent memoir on the maUceuvering
of ships.

A considerable share of his attention was also employed on the subject of achromatic telescopes,
with the different kinds and combinations of glass, to render their effects colourless. These labours
are contained in three memoirs printed by the academy in the years 1756, 1757, and 1762. But it
were endless to particularise all his valuable labours.

This last year, however, terminated the useful existence of this great man ; his death being occa-
sioned by a severe cold he had taken, in returning home one night after supper, at 53 years of age.



TOL. XL.] PHILOSOPHIC AI. TSANSACTIONS. ill



the rotation of NnmM will be -(2ru — uu)du, where c is the circumference

to the radius r.

Now because of the smallness of nm, we may account all the particles of
matter contained in that space as equally attracting the corpuscle at a ; there-
fore we shall have the attraction of that small space, if we multiply that solidity

• 1 A P

by the attraction at m, and that attraction at m is — ; X — . Thus will be had

-' ' AM* AM

analytically ^^"^^ .j,2ru—uu.du = j^r^Tsr^^''''" ^ " ~ udu^u) the in-
tegral of which, - — — (4-r«v^« — 4-w"v/w)> is the attraction of the space
arising from the rotation of anm. In which value if we take u = 2r, we have
by reduction -^ac. Hence the attraction of the whole space aeqc is expressed ;
then adding ^ for the attraction of the whole sphere, we have -j-c -f- -^c for
the attraction of the ellipsoid.

Carol. — For an oblong spheroid a will be negative, and the united attraction
will be -I* — tVc.

Note. — If the foregoing spheroid, instead of circular elements arising in pn,
consist of other elements, for instance elliptical, which should differ from a '
circle no more than the ellipsis ae, and by which there would be the same super-
ficies as by the circles pn, it appears that the attraction will be still the same,
because in those elements pn, whatever the remaining force might be, the circles
PM being taken away, it will be as it were composed of parts which would have
the same attraction as on that of the ellipsoid, having regard to the smallness
of NM, and to the equable quantity of matter.

Lemma. Let kl be a circle, fig. 1 3, h its centre, vh perpendicular to the
area of the circle, and nh = vh, but making with it an angle infinitely small,
or very small: then the attraction of the circle kl on n, may be taken, without
sensible error, as the attraction of the circle on v; or, which is the same thing,
that the one attraction differs from the other only by a quantity infinitely less,
with respect to both, than as vn is less in respect to hv.

To demonstrate which proposition, it must be shown, that two corpuscles
being placed at the extremity of any diameter kl, there is one attractive force
at N, and another force at v, the sum of which may be accounted the same.
But, neglecting the computation for having the attraction of the body at k on
the corpuscle at n, it will be easily seen that it will be the same with the attrac-
tion on V, to which should be added a small quantity involving nv. In like
manner it may be seen, that the attraction of a body at l on the corpuscle n,
will be the same as the attraction on v, deducting the same small quantity.
Therefore the sum of both these attractions is one and the same.

VOL. VIII. R



1^22 PHILOSOPHICAL TRANSACTIONS. [aNNOI737.

Corol. — Instead of the circle kl, if there was a certain ellipsis, or any other
curve line, which should differ very little from a circle, by the same arguments
as in the note, it is easily gathered that the foregoing proposition would always
hold good.

Theorem I. Let AEae, fig. 14, be an elliptic spheroid, the axis of revolu-
tion being Art ; then the attraction this spheroid exerts on a corpuscle at n, is
the same as that attraction, which any spheroid exerts whose pole is n, its axis
of revolution ntz, and its second axis the radius of a circle, having the same
superficies as fg, the elliptic section of the ellipsoid AEae, by a plane erected
perpendicularly on fg, its conjugate diameter.

To -demonstrate this, conceive innumerable elements kl, parallel to the
ellipsis FG, that is, all erected on ordinates to the diameter, it is evident that
the spheroid AEae will differ from the aforesaid spheroid only in this, that in
the first all the elements make with cn an angle differing infinitely little from
a right angle, but in the second all the elements make a right angle, without
any difference, while in both spheroids the elements have the same superficies.
But, by the preceding proposition, the attraction of every element kl on n, is
considered as the same in both cases ; but as to the thickness of the elements
kA/l, we may take hA for the perpendicular hi, because of the smallness of the
angle jAh ; therefore the total attraction of both spheroids may be taken the
one instead of the other.

Prob. II. To find the attraction of the spheroid AEae on a corpuscle at any

point N.

Let AC = a, CE = b, cn = r, CG the conjugate diameter to on will be

- since a and b have very little difference; from the preceding proposition,

find the attraction of the spheroid, whose greater axis is r, and the less

.abb 7 ,a

\/— or by/ J.

For this, we must apply the formula which we found in prob. 1, viz. ^c —

tVJC, or ^pr rt^pr, putting pr for c, but in this formula instead of a sub-

stituting '■-~^'^~ = 1 — ^ /p or \n — 7n, putting a-\-ma for b, and a -j- Tza
for r and in the computation neglecting the second powers of n and m.

If therefore \n — mhe. put instead of a, the aforesaid formula will become
^pr _ ^prn -\- -J^prm, or fjba — -^pan + -^pam ; which expression is the re-
quired attraction of the spheroid on n.

If n = 0, then we have ^pa + -^pam for the attraction on the pole a.

But if n = m, then we have \pa -f- -hpa^ for the attraction at the equator.

Theorem II. Let AEae, fig. 12, as before, be a spheroid, whose axes differ



VOL. XL.] FHILOSOPHICAL TRANSACTIONS. 123

by a very small quantity, which for greater perspicuity I shall call infinitely
small. If tins spheroid be conceived to be of a fluid and homogeneous matter,
and revolved about the axis ha in correspondent time, that the gravity of the
column CE may be equal to the gravity of the column ac, that is, by the New-
tonian principles, the attraction in e, neglecting the centrifugal force, may be
to the attraction at a, as ca to ce : I say, that all the columns on, wanting an
infinitely small quantity of the second degree, will preserve an equilibrium with
those two columns; that is, the attraction on n, neglecting the centrifugal
force, simply in the direction on, is to the attraction on a, as ca to on.

For the demonstration, the same notation will serve as in the preceding pro-
position: first find the centrifugal force at e, which may agree with the equili-
brium of the columns ce, ca. Then say, as \pa -)- -^pam — f: fpa -\- -J^pam
:: 1 : 1 -|- »n, hence is found/= -^pam.

Then for exhibiting the gravity at n composed of the attraction, omitting
the centrifugal force, find the centrifugal force at n, or, which is the same,
on M upon the sphere, which must diff^er from each other only by an infinitely
small quantity of the second order, if de be supposed to express the centrifugal
force y at e, then mn will express the centrifugal force at n, for the centrifugal
forces are as the radii, when the times of revolution are the same, and by the
property of the ellipses it is de : nm :: cb : MP.

But if the centrifugal force act in the direction np, it must be reduced to nc,
and NO will be the remaining part. Therefore the centrifugal force at n, or at
M, is to the centrifugal force at e, or at d, as no is to de. Therefore the ex-
pression for the centrifugal force at n will he'-Jt^pan, and consequently the ex-
pression for the gravity there will be ^pa — -^pan -{• -^pam — -^pan, or \pa
— ^pan -\- -^pam.

Now to find the centrifugal force at n, which results from the equilibrium of
the columns, the gravity at a must be to the gravity at n, as nc to ac; but the
gravity at a is ^pa -f- -^pam, which expression being drawn into or J — n,

after reduction it becomes ^pa — ^pn -|- -^pam, and is the same expression as
that above.

Hence we see that there can be only an infinitely small difference between
the figure which the earth ought to have by the Newtonian hypothesis, and
the ellipsoid. For as the quantity de is about the 230th part of ac, in the
preceding computation we neglect only a quantity of the same order with






124 PHILOSOPHICAL TRANSACTIONS. [^ANNO 1737.

On the Efficacy of Oil of Olives in curing the Bite of Vipers. By Stephen
miliams, MB. F.R.S. N" 445, p. 26.

In presence of several gentlemen of the faculty of physic, Wm. Oliver, the
viper-catcher (mentioned N° 443,*) suffered himself, on June 26, 1735, to be
bitten by a female viper ; which being enraged, fixed her fangs in the middle
part of his fore-finger. Blood soon issued out at the wounds : but that the
poison might more strongly appear, the same viper immediately bit a pigeon in
the breast, which expired in less than half an hour. Another pigeon was also
bitten by the same viper, which expired also, though not so soon as the first. The
man immediately complained of an acute pain in the wounded part ; and it soon
looked red, then became of a livid colour : his finger swelled to a great size,
and he could not bend it. Soon after this his hand also began to swell : he
complained of faintness, aftd pains flying to his arm, shoulder, and arm-pits.
In half an hour's time from the bite, his specific being applied, and strongly
rubbed into the part aflfected, procured him immediate ease. His pain lessened,
his finger became flexible, his spirits seemed more chearful : the specific being
thus several times repeated and applied, his pains gradually diminished. The
next day his finger and hand remained swoln, but without pain : the skin began
to appear yellow, and pustules like bladders appeared on his finger ; which being
pricked, emitted a sanious liquor. In 2 days time all his symptoms vanished,
and he became perfectly well.

June 30, the gentlemen of the faculty met again, and tried several experi-
ments on puppies, cats, and pigeons ; when they proved the efficacy of this
man's specific, to the great satisfaction of the company.

J4 Proposal for the Measurement of the Earth in Russia. By Mr. Jos.
Nic. de Vlsle, first Professor of Astronomy, at Petersburg, and F.R.S.
N° 445, p. 27.

Necessity, or the exigencies of geography and navigation, put mankind very
early on measuring the earth. On the first determination of the magnitude of
the earth in geographical measures, as in Stadia and Arabian miles, the ancients
did not employ any great degree of exactness. They were content to set down
the circumference of the earth, and of its parts, in round numbers ; probably
because they did not expect to be able to attain much preciseness in a research
of this nature. But as their desires of improving geography increased, they

• Page 84, of this vol. of these Abridgments.



VOL. XL.} PHILOSOPHICAL TRANSACTIONS. 12S

found it necessary to have a more exact knowledge of the magnitude of each
degree, not only in great measures, as in miles and leagues, but also in perches,
toises and feet.

As to Russia, the geographical measures of which are wersts, divided each
into 500 sagenes, and each sagene supposed to be exactly 7 feet English ; this
proportion once known, and the exact ratio of the English to the French foot,
or to the toise of 6 feet, which the French astronomers employed in their
measurements, and of which they found a degree of a great circle contained
57060 ; if it be asked what more is requisite for concluding that a degree of a
great circle contains 104-1- wersts ? and what remains towards the perfection of
the geography of Russia, in the most minute detail, but to employ this mea-
sure of wersts, sagenes, and English feet, in actual measurements; and to con-
struct the charts by the most exact methods of geometry ; it may be answered,
we should be very happy, if in the geography of Russia we were arrived at this
pitch ; not only in the general map, but in that of any particular district, the
nearest and of most concern to us. But besides that we are as yet far from pre-
tending to this ; it may be made to appear, that it is not possible to attain it,
without undertaking an equal, and even a greater work than all that has been
hitherto done in France and elsewhere, towards the measurement of the
earth.

For if the earth be not truly spherical, all the degrees of the great circles will
not be equal to each other ; and those of the small circles, taken at a certain
distance from their parallel great circles, will not have the same relation that
the degrees of the smill circles, taken at the same distance, would have on a
sphere. In all this there might possibly arise an infinite variety, according to
the figure the earth might have; and as it is not yet decided what is the earth's
true figure, and that there is no better method of ascertaining it than by obser-
vations made in so great an extent as that of Russia; therefore the perfection'
(of the geography of Russia stands in need of this great undertaking ; which,
besides its usefulness, will yield much honour, by contributing towards the de-
ciding the celebrated question of the earth's figure.

There have been some who have long since suspected, and even thought
they were furnished with proofs, that the earth is not exactly spherical. But
supposing the earth to be bounded by a curve surface, such as it would be by
the level of the sea carried quite over all the earth; it is in this manner, the
earth being considered as covered with a fluid, that Sir Isaac j^ewton, in the
first edition of his Principia, has demonstrated, that supposing this fluid homo-
geneous, and the earth to have been at rest at the time of its creation, it must
have assumed the figure of a perfect sphere: but afterwards, supposing it to



I2(J PHILOSOPHICAL TRANSACTIONS. [ANNO 1737.

have a motion on its axis, as is well known it has in 24 hours ; this spherical
figure must have been changed into that of a spheroid, flatted at its poles, in
which the degrees on the meridian must be greater towards the poles, than near
the equator.

Sir Isaac confirms this hypothesis of the earth's figure, by observations of the
diminution of the simple pendulum on approaching the equator : to which Dr.
Pound adds the analogy the earth has with some of the other planets, as Jupiter,
which sometimes appears oval, its least axis being that about which it makes its
revolution.

This opinion of Sir Isaac has likewise been maintained by Mr. Huygens,
though with some small difference. But in J 691, Mr. Eisenschmid having
compared the measurements of the earth made in different latitudes, as that of
Father Riccioli in Italy, of Mr. Picart in France, and of Snell in Holland; and
having found that the degree, which resulted from those different measure-
ments, continued to decrease in approaching the poles, which is quite the con-
trary of what follows from the earth's figure supposed by Sir Isaac and Huygens,
Mr. Eisenschmid was therefore of opinion, that the earth was longer at the
poles.

This opinion of Mr. Eisenschmid was afterwards confirmed by the late Mons.
Cassini, in the observations of the meridian of Paris. For in 1701, having
carried on these operations to the Pyrensean mountains, which is a space of
above 7+ degrees, he found that as he advanced to the south, these degrees in-
creased -5-i-5^ part, or Tl toises each degree.

Since the meridian of Paris was, in 1 7 1 8, carried on northward to the sea,
M. Cassini, the son, found on comparing more than 8 degrees, which this
meridian contains from sea to sea, that the increase, going northward, was but
from 60 to 61 toises each degree; as may be seen in the large treatise published
in a separate volume, as a sequel to the memoirs of the Royal Academy of
Sciences of Paris for the year 17I8. These reasons did not hinder Sir Isaac
from persisting in his first opinion of the figure of the earth flatted at the poles, as
appears in the 2d and 3d editions of his Principia, published in 1713 and 1726:
and it is very surprising, that by this very figure of the earth he demonstrates a
certain motion it has, to explain in the Copernican system the precession of the
equinoxes, or the apparent motion of the fixed stars in longitude. Sir Isaac
finds the inequality of the degrees on the meridian, in so little an extent as that
of France, not sensible enough to be possibly determined by immediate obser-
vations ; and he is of opinion, that we ought more to rely on the observations
of the simple pendulum, and on the other principles which he has built upon,
to conclude the earth flatted at the poles.



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. ]"27

In 1720, M. Mairan attempted to reconcile the two different hypotheses of
Sir Isaac and M. Cassini, by imagining that the earth, at its creation, being
without motion, was of a much more oblong figure than that which Cassini



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