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The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) online

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of elliptic sectors, and particularly to some general theorems concerning the
multiplication and division of circular sectors or arcs. When 2 points are as-
sumed in an hyperbola, and also in an ellipsis, so that the sectors terminated
by the semi-axis, and the 2 semi-diameters, belonging to those points, are in
the same given ratio in both figures, then the relation between the semi-axis
and the 2 ordinates drawn from those points to the other axis, is always defined
by the same, or by a similar equation in both figures. This proposition serves
for demonstrating Mr. Cotes's celebrated theorem, as it is extended by M. De
Moivre, by which a binomial or trinomial is resolved into its quadratic divisors,
and various fluents are reduced to circular arcs and logarithms. The demon-
strations are also rendered more easy of the theorems concerning the resolution
of a fraction, that has a multinomial denominator, into fractions that have the
simple or quadratic divisors of the multinomial for their several denominators.
These demonstrations are derived from the method of fluxions itself, without
any foreign aid; the invariable coefiicients being determined by supposing the
variable quantities or its fluxions to vanish.

When a fluent cannot be assigned by the areas of conic sections, it may
however be measured by their arcs in some cases; and this may be considered
as the 3d degree of resolution, or the fluents may be called of the 3d order.
On this occasion some fluents are found to depend on the rectification of the
hyperbola and ellipsis, which have been formerly esteemed of a higher kind.
The construction of the elastic curve, with its rectification, and the measure
of the time of descent in an arch of a circle, are derived from hyperbolic and
elliptic arcs; and the fluents of this kind are compared with those of the first
or second order by infinite series. Because there are fluents of higher kinds
than these, the trajectories above-mentioned, which are described by a centri-
petal force, that is as some power of the distance from a given centre, when
the velocity of the projection is that which would be acquired by an infinite
descent, or by such a centrifugal force, and the velocity is such as would be
acquired by flying from the centre, are employed for representing them. A
simple construction of these trajectories had been given above, by drawing rays


from the centre to a right line given in position, increasing or diminishing the
logarithms of those rays always in a given ratio, and increasing or diminishing
the angles contained by them and the perpendicular in the same ratio. From
any figure of this kind a series of figures is derived by determining the inter-
sections of the tangents of the figure with the perpendiculars from the centre.
Every series of this kind gives 2 distinct sort of fluents; and any one fluent
being given, all the other fluents taken alternately from it in the series depend
on it, or are measured by it ; but it does not appear that ihe fluents of one
sort can be compared with those of the other sort, or with those of any dif-
ferent series of this kind.

The inverse method is prosecuted farther in the 4th chapter, by various
theorems concerning the area when the ordinate is expressed by a fluent, or
when the ordinate and base are both expressed by fluents. The first is the J 1th
proposition of Sir Isaac Newton's Treatise of Quadratures. In art. 81 9, 820,
&c. the author supposes the ordinate and base to be both expressed by fluents,
and shows, in many cases, that the area may be assigned by the product of two
simple fluents, as of two circular arcs, or of a circular arc and a logarithm.
This subject deserves to be prosecuted, because the resolution of problems is
rendered more accurate and simple, by reducing fluents to the products of
fluents already known, than by having immediately recourse to infinite series.
One of the examples in art. 822 may be easily applied for demonstrating, that
the sum of the fractions which have unit for their common numerator, and the
squares of the numbers 1, 2, 3, 4, 5, 6, &c. in their natural order, for their
successive denominators, is -^ part of the number, which expresses the ratio of
the square of the periphery of a circle to the square of its diameter; which is
deduced by Mr. Euler, Comment. Petropol. tom. 7, in a different manner,
and other theorems of this kind may be demonstrated from the same or like

The series that is deduced by the usual methods for computing the area or
fluent, converge in some cases at so slow a rate, as to be of little or no use
without some farther artifice. For example: the sum of the first 1000 terms
of Lord Brounker's series, for the logarithm of 2, is deficient in the 3th deci-
mal. In order therefore to render the account of the inverse method more
complete, the author shows how this may be remedied, in many cases, by
theorems derived from the method of fluxions itself, which likewise serve for
approximating readily to the values of progressions, and for resolving problems
that are commonly referred to other methods. Those theorems had been de-
scribed in the first book, art. 352, &c. but the demonstration and examples
were referred to this place, as requiring a good deal of computation. The base


being supposed equal to unit, and its fluxion also equal to unit, let half the
sum of the extreme ordinates be represented by a, the difference of the first
fluxions of these ordinates by b, the difference of their 3d, 5th, 7th and higher
alternate fluxions by c, d, e, &c. then the area shall be equal to

'^ - ll + 7lo - 35iio + lisieoo - ^""^ ^^'^h '' ^h« ^''^ th^°'*«'" ^°'" fi"d'"&
the area. The rest remaining, let a now represent the middle ordinate, and

the area shall be equal a + A _ _|i_ + ^^ _ _^_ + &c. And this
is the theorem which the author makes most use of. When the several inter-
mediate ordinates represent the terms of a progression, the area is computed
from their sum, or conversely their sum is derived from the area, by theorems
that easily flow from these.

These general theorems are afterwards applied for finding the sums of the
powers of any terms in arithmetical progression, whether the exponents of the
powers be positive or negative, and for finding the sums of their logarithms,
and thereby determining the ratio of the uncia of the middle term of a bino-
mial of a very high power to the sum of all the uncia;. This last problem was
celebrated among mathematicians some years ago, and by endeavouring to re-
solve it by the method of fluxions, the author found those theorems, which
give the same conclusions that are derived from other methods. They are
likewise applied for computing areas nearly, from a few equidistant ordinates,
and for interpolating the intermediate terms of a series, when the nature of
the figure can be determined, whose ordinates are as the differences of the

In the last chapter, the general rules, derived from the method of fluxions
for the resolution of problems, are described and illustrated by examples.
After the common theorems concerning tangents, the rules for determining
the greatest and least ordinates, with the points of contrary flexure, and the
precautions that are necessary to render them accurate and general, which were
described above, are again demonstrated. Next follow the algebraic rules for
finding the centre of curvature, and determining the caustics by reflexion and
refraction, and the centripetal forces. The construction of the trajectory is
given, which is described by a force that is inversely as the 5th power of the
distance from the centre, because this construction requires hyperbolic and
elliptic arcs, and because a remarkable circumstance takes place in this case,
and indeed in an infinity of other cases, which could not obtain in those that
have been already constructed by others, viz. that a body may continually de-
scend in a spiral line towards the centre, and yet never approach so near to it
as to descend to a circle of a certain radius; and a body may recede for ever



from the centre, and yet never arise to a certain finite altitude. The construc-
tion of the cases wherein this obtains, is performed by logarithms or hyper-
bolic areas, the angles described about the centre being always proportional to
the hyperbolic sectors, while the distances from the centre are directly or in-
versely as the tangents of the hyperbola at its vertex. The circle is an asymp-
tote to the spiral ; and this can never be, unless the velocities requisite to carry
bodies in circles increase while the distances decrease, or decrease while the
distances increase, in a higher proportion than the velocity in the trajectory;
that is, unless the force be inversely as a higher power of the distance than the
cube. Next follow theorems for computing the time of descent in any arc of
a curve, for finding the resistance and density of the medium, when the trajec-
tory and centripetal force are given, and for defining the catenaria and line of
swiftest descent in any hypothesis of gravity.

Then the usual rules are derived from the inverse method for computing the
area, the solid generated by it, the arc of the curve, and the surface described
by it, revolving about a given axis. The meridional parts in a sphere, and any
spheroid, are determined with the same accuracy, and almost equal facility.
The attraction of a spheroid at the equator, as well as at the poles, is deter-
mined in a more general manner than in the first book, or in a piece of the
author's, published at Paris in 17-10, which obtained a part of the prize pro-
posed by the Royal Academy of Sciences for that year. Several mechanical
problems are resolved, concerning the proportion the power ought to bear to
the weight, that the engine may produce the greatest efi^ect in a given time;
and concerning the most advantageous position of a plane which moves parallel
to itself, that a stream of air or water may impel it with the greatest force,
having regard to the velocity which the plane may have already acquired. On
this occasion, it is shown, that the wind ought to strike the sails of a wind-mill
in a greater angle than that of 54° 44', against what has been deduced from the
same principles by a learned author. The same theory is applied to the motion
of ships, abstracting from the lee-way, but having regard to the velocity of the
ship; and among other conclusions it appears, that the velocity of a vessel of
one sail may be greater with a side-wind, than when she sails directly before
the wind ; which, perhaps, may be the case of those seen by Captain Dampier
in the Ladrone Islands, that sailed at the rate of 1*2 miles in half an hour with
a side-wind.

The remainder of this chapter is employed in reducing equations from second
to first fluxions; constructing the elastic curve by the rectification of the equi-
lateral hyperbola; determining the vibrations of musical chords; resolving pro-
blems concerning the maxima and minima, that are proposed with limitations,

VOL. vui. 4 R


relating to the perimeter of the figure, its area, the solid generated by this
area, &c. with examples of this kind concerning the solid of least resistance ;
and concludes with an instance of the theorems by which the value of the ordi-
nate may be determined from the value of the area, by common algebra, and by
observing, that it is not absolute, but relative space and motion, that is supposed
in the method of fluxions.

Observations on the Mouth of the Eels* in Vinegar, and also a strange j4quatic
Animal ; in a Letter from the Rev. Mr. Henry Miles to Mr. Baker, F. R. S.
■^ N° 469, p. 416.

This observation was made with the camera obscura microscope : first, in a
very small tube, Mr, M. put a small quantity of vinegar, with several anguillae:
at first sight of the image on the screen, one had a motion as if it had been
wounded, about the middle of the back ; it neither rose nor sunk in the liquor,
but lay in this form -^==5*-^ , wriggling itself, as if giving signs of pain, and would
soon expire, which it accordingly did in a minute's time ; but it coiled itself up,
and stuck to the side of the tube very close. He put out the liquor, after wait-
ing to see whether it would revive, in vain, and viewed it several times in the
common light, in which way he had the most distinct appearance. The larger
end, which may be called the head, was stretched out from the rest of the body,
a little way, as in the figure, which gave an opportunity of examining what
mouth it had. On the first view of it in common light, he saw what he thought
may be called the mouth : repeated trials in difli^erent lights and positions, and
with different magnifiers, confirmed the suspicion. After the strictest and most
exact observation he could discern it to be nothing more than a transparent
tube. Where the instruments of nutrition, and the springs of life, are, he
doubts not we shall soon discover.

In this figure, a is the mouth, which seemed to be as wide open as
' a it possibly could be. The figure is too small to give a just idea of the
shape of the mouth, but it had the appearance which a tube, or
rather a cone, would make cut slopewise.

: Mr. H. Miles also sent some specimens of an odd aquatic animal,-^ found in
standing water : he kept some of them in their own element in the house, but
they all died in a day and half's time. They seem to be nothing but skin, and

• The vibrio aceti, Linn. Gmel. is sometimes considered as a variety only of the vibrio glutinis or
paste eel, from which it chiefly differs in being more slender and transparent.

+ The supposed animal here described by Mr. Miles, is nothing more than the seed of the plant
called bidens tripartita, which usually grows in watery places, and the seeds of which have a kind of
elastic motion when touched.


seem no thicker when alive: they have the power, as most aquatic insects have,
of sinking to the bottom on the approach of a stick, &c, and fall like a piece of
rotten wood, or leaf. When taken out of the water, if laid on a paper, &c.
they will spring away like a grashopper.

P. S. The animals were caught the day before, and kept in water in a glass; and
when he had finished his letter, and went to pack them up in paper, he found
none of them left, as he thought, at first; but on a nearer view he found they
were all collected together in a knot, which he took for some filth in the water,
till he more carefully viewed them, and found them hanging together by the tails.

An Extract from the Books of the Town-Council of Edinburgh, relating to a
Disease there, supposed to be Venereal, in the Year 14^7, in a Letter from
Mr. Macky, to Mr. Maclaurin. N° 469, p. 420.

If the venereal disease was never known in Europe till the siege of Naples
1495, it must have made a very quick progress indeed through Europe; for
in 1497, I find it raging in Edinburgh, and the king and his council terribly
alarmed at this contagious distemper, as appears from a proclamation of King
James the 4th, in the records of the Town-Council of Edinburgh. The minute
of council is dated the 22d of September. I have taken a copy of it for your
amusement, and, if you please, you may communicate it to the Society. I have
pretty nearly observed the old spelling, except in numbers.

Sept. 11, 1497. — " It is our Soverane Lords will and the command of the
Lordis of his Counsale send to the Provest and Baillies within this bur' that this
proclamation foUowand be put till execution for the eschewing of the greit ap-
pearand danger of the infection of his leiges fra this contagious sickness callit
the grandgor and the greit uther skayth that may occur to his leiges and in-
habitans within this bur' ; that is to say, we charge straitly and commands be
the auhority above writtin, that all manner of personis being within the free-
dom of this bur' quilks are infectit or hes been infectit uncurit with this said
contagious plage callit the grandgor, devoyd, red and pass fur' of this town and
compeir apon the sandis of Leith at ten hours before none and thair sail thai
have and fynd botis reddie in the havin ordanit to them be the officeris of this
bur' reddely furneist with victuals to have thame to the Inche,* and thair to
remane quhill God proviyd for thair health : and that all uther personis the quilks
taks upon thame to hale the said contagious infirmitie and taks the cure thairof
that they devoyd and pass with thame sua that nane of thair personis quhilks
taks sic cure upon thame use the samyn cure within this bur' in pns nor peir

* An island in the Frith of Edinburgh over against Leith.


any manner of way. And wha sa beis foundin iiifectit and not passand to tlie
Inche as said is be Mononday at the Sone ganging to, and in lykways the said
personis that takis the sd cure of sanitie upon thamegif they will use thesainyn
thai and ilk ane of thame salle be brynt on the cheik with the marking irne that
thai may be kennit in tym to cum and thairafter gif any of tham remanis that
thai sail be banist but favors."

Some Account of the Insect* called the Fresh- water Polypus, beforementioned in
these Transactions, N" 466 and 467. By Martin Folkes, Esq. Pres. R. S.
N" 469, p. 422.

After several experiments and observations on these animals, much of the
nature of those in the papers referred to in N°* 466 and 467, which it is not
necessary here to repeat, Mr. Folkes gives a description of several magnified
views of the animal, as referring to the several states of it, described in the
paper, as follows :

PI. 17, fig. 1 represents a polypus as seen in the microscope, when in a
state of extension, the arms spread as when feeling for their prey, and the
mouth sharp and prominent. Fig. 2 and 3 represent the same insect in its most
contracted state. Fig. 4 and 5 show the insect when in a middle state of con-
traction ; the body is then wrinkled, so as to appear somewhat like a grub or
earth-worm. Fig. 6 is a polypus with a young one growing from its side, and
another from that again : this is not so much extended as that in fig. 1, and is
to be supposed to have taken lately some food, by which the cavity of the inside
is made more conspicuous, and the communication of the guts of the young
ones with those of the parents becomes sensible. Fig. 7 shows the appearance
of a polypus, that has already swallowed the best part of a worm endwise. He
is grasping the remaining part to draw that in also. Fig. 8 represents a polypus,
whose mouth is greatly extended : he has just taken in the middle part of a
worm ; the opening of the mouth is there remarkable, the arms seem some-
what contracted from the effort in stretching the mouth so wide ; the neck also
may be there observed between the mouth and the stomach, but which will
soon disappear as the worm is sucked further in. Fig. 9 is another polypus,
nearly in the same state as the last ; but the worm is omitted in the figure, to
show the form of the mouth more distinctly. Fig. 10 shows the same polypus
when the worm is drawn quite double into his stomach ; here the neck entirely
disappears, and the whole is like an open bag or purse. Fig. 1 1 is the same
polypus, after he has entirely swallowed his worm ; the mouth is now again

* Zoophyte ; see the remark subjoined to Dr. Mortimer's paper, p, 623 of this vol. of these


closed and contracted, and the worm may be discovered through the skin, as it
lies coiled in his stomach. In these last 5 figures it may be noted, that, how-
ever extended and swelled the stomach of the insect appears, the posterior part
is not stretched in proportion, but discovers itself every where as a small tail, in
which is contained a gut, with which the stomach communicates. Fig. 12
shows one of the horns or arms of a polypus very much magnified, for giving
some imperfect idea of the knots or papillas in the transparent membranous
substance, of which it is composed. Fig. 13 represents a polypus that had
several young growing from it at once, some of which had also others springing
out from them again. Besides those here represented, 8 other young ones
had at several times separated themselves from him, since the insects were

yin Account * of a Book intitled, New Principles of Gunnery, containing the
Determination of the Force of Gunpowder ; and an Investigation of the re-
sisting Power of the Air to swift and slow Motions. By B, R. F. R. S. as far
as the same relates to the Force of Gunpowder. N° 469, p. 437.

This treatise contains 1 chapters. The first treats of the force of gunpowder,
and the velocities communicated to bullets by its explosion : the 2d considers
the resistance of the air to bullets and shells moving with great velocities ; and
endeavours to evince, that this resistance is much beyond what it is generally
esteemed to be ; and consequently that the track described by the flight of these
projectiles, is very different from what is usually supposed by the modern writers
on this subject.

The principal points endeavoured to be established in the first chapter are
these, " That the force of fired gunpowder is no more than the action of a
permanent elastic fluid, which is produced by the explosion; that this fluid ob-
serves the same laws with common air in their exertion of its pressure or elasti-
city ;" and consequently, " That the velocities communicated to bullets by the
explosion may be easily computed from the common rules, which are established
for the determination of the air's elasticity."

The first 2 propositions contain the proofs that a permanent elastic fluid is
constantly generated in the explosion of gunpowder. The 3d proposition is,
that the elasticity of this fluid produced by the firing of gunpowder, is, caeteris
paribus, directly as its density. This is proved both by the author's own ex-
periments, and by several of Mr. Hauksbee's ; when, by the firing of 26 quan-
tities of powder successively, the mercurial gauge was sunk from 294^ inches, to

* This account was given by the very ingenious author himself, Mr. Benjamin Robins.


124- ; for by comparing these experiments together, and making the necessary
allowances, it will be found, that the elasticity was nearly proportional to the
density in all that variety of densities.

In this proposition, the analogy between the fluid produced by the explosion
of powder and common air, is established thus far, that they exert equal elasti-
cities in like circumstances; for this variation of the elasticity, in proportion to
the density, is a well-known property of common air. But other authors, who,
since the time of Mr. Boyle, have examined the factitious elastic fluids pro-
duced by burning, distillation, &c. have carried this analogy much further, and
have supposed these fluids to be real air, endued with all the properties of that
we breathe; particularly Dr. Hales, who has pursued this examination with the
greatest exactness, in a series of the best contrived processes, constantly affixes
the denomination of air to these factitious fluids, he having found that their
weight is the same with that of common air, and that they dilate with heat,
and contract with cold ; and that they vary their densities, under different de-
grees of impression, in the same proportion with common air; and from hence,
and other circumstances of agreement between them, he supposes them to be
of the same nature with air, and conceives them to be fitly designed by the
same name.

But so perfect a congruity between these factitious fluids and air, is not
necessary for the purposes of this treatise. The fundamental positions of this
first chapter supposing no more, than that the elasticity of the fluid, produced
in the explosion of gunpowder, is always, caeteris paribus, as its density ; and
that the force of fired gunpowder is only the action of that fluid modified ac-
cording to this law.

The law of the action of this fluid being determined, 2 methods offer them-
selves for investigating the absolute force of powder on the bodies it impels be-
fore it. The first by examining the quantity of this fluid produced by a given
quantity of powder, and thence finding its elasticity at the instant of the ex-

Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 78 of 85)