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a sufficient proof of his general calculations.

Mr. Kersseboom then gives an account how many people were buried in the
city of Dort every year, from 170O to 1739 inclusive, amounting, in 40 years,
to 28977 persons ; which is annually, on an average, 724. — ^The marriages are
202 couple annually, during the same time, which should produce (according
to the author's calculations in his first treatise) 325 children per 100 marriages,
and consequently 656 children per annum ; but he has found it, on an average,
to be 651. — This city being a sea-port, and driving a large trade to Scotland,
and on the Rhine, and consequently many of the people, whose traffic brings
them to Dort, may die there, it is supposed, that about 680 children are bor»
annually there, and that consequently this city may contain 24000 souls.

Next to this, the author gives an account of Haerlem, how many people died
there in 84 years, from 1656 to 17 39 inclusive, namely, 132 1 32 persons,
which is annvially, on an average, 1573. — The next is, how many marriages, from
anno 169O to I739 inclusive, namely, 2191O, is annually 438, on an average.
— As to the births, Mr. Kersseboom refers to his first treatise, p. 54, where he
supposes, that 1450 children may be born alive annually; and endeavours to
demonstrate it further, by giving an account of the births for 60 years, namely,
from 168O to 1739, and finds it to be 1453 ; from which it is calculated, that
this city contains 50500 souls, as mentioned in his first treatise.

The next account is that of the burials of Delft and Delftshaven, from the
years 1724 to 1739, being 15 years, and is found to be annually, on an average,
723 persons ; but there is subjoined, for the greater certainty, an account from
the year 1696 to 1739, which proves it to be 748 persons annually.

The marriages are, in the same time of 44 years, on an average, 224 per
annum, which should produce 728 children, according to the rule laid down
before, namely, 100 marriages producing 325 children; but is found to produce
from 1690 to 1739 inclusive, to be 648 per annum, on an average; from
whence it is supposed those two places contain 25000 souls.

The city of Leyden comes next in consideration. It appears by a list for 50
years, namely, from the year 1690 to 1739 inclusive, that there have been
buried in that city annually, on an average, 1919 persons ; and married during
the same time, annually, on an average, 558 couple, which, agreeable to the
former rule, would produce 1813 children per annum, but is found to have
been 1834 per annum, on a medium, as aforesaid ; the author concludes con-
sequently, that this city contains 63000 souls.

The next city is Amsterdam : it appears by a list, that since the year i6q6 to



630 PHILOSOPHICAL TRANSACTIONS. [anNO 1742-3.

1738 inclusive, there have been buried in this city 7323 persons annually (Jews
excepted) ; and there having been married, during the same time of 43 years,
2311 couple annually, produced, according to the author's computation, 7134
children annually, at a medium ; and takes it thence for certain, that Amster-
dam contains (including 20000 Jews, as observed in his first treatise, p. 21)
241000 souls. — The author proceeds, in the like manner, about other places.
He then gives a table showing how long 432 widows lived during a century,
and finds it to have been near 14 years each on a medium ; and then subjoins
a list how many years married people of different ages continue to live probably
together, before the bonds of matrimony, by the death of either party, are dis-
solved ; namely,

live between

Those whose ages together are 40 . . 24 and 25 years.

30 . . 22 and 23.

60 . . 20 and 21.

70.. 19and20.

80.. 17 and 18.

go.. 14 and 15.

100.. 12 and 13.

And finishes with rejecting the method of calculating the quantity of people
after the manner of Vossius, Auzout, Petty, and others.

The third treatise contains, 1 st, A copy of a letter written by the author
in the beginning of the year 1741, to Mr. John Eames, F. R. S. 2dly, A de-
monstration, in 29 tables, that Mr. Simpson's calculation of lives, as 1 to 26,
is a mistake, and his own hypothesis, as 1 to 35, right ; and proves, from Mr.
Maitland's Observations, that children in London, of 2 years old, continue to
live, on a medium, above 37 years ; and observes, that Dr. Halley's table has it
full 38 years and a half.

The author supposes, 3dly, That out of every lOO children born, 5 come
dead into the world; and that out of every 100 children born alive, near 20
die under a year old ; and he shows, 4thly, how much Mr. Simpson differs in
his calculation; namely. That full 32, out of 100 children, die under a
year old.

The rest of this treatise consists in divers calculations and tables of interest,
and the value of annuities for life on different ages and interest ; and concludes
with an explanation of the same, and the usefulness thereof.



VOL. XLI/.J PHILOSOPHICAL TRANSACTIONS. 631

Concerning the wonderful Increase of the Seeds of Plants, e. g. of the Upright
Mallow. By Mr. Joseph Hobson of Macclesfield. N° 4t)8, p. 320.

In the upright mallow, the seeds being disposed in rings, Mr H. counted
those which were on the principal stems, and found them as follows.

Rings in all lOlQQ

Multiply by seeds in one ring 12 Seeds.

Number of seeds . 1 22388

Allow for two large stems destroyed 7612

Seeds in all J 300OO



He then counted the seeds in several particular rings,, and found them com-
monly 14 in each, but has confined himself to multiply the rings by 12, which
is moderate, yet makes the number of seeds amount to 130000, allowing 7612
seeds for 2 large stems cut down and destroyed, a moderate allowance, con-
sidering 2 of the stems alone contain each above 1000 rings: some of these
stems were above 2 yards and a half high. This plant was a seedling last year,
transplanted out of the fields on the end of a sloping strawberry-bed ; and he
counted the rings in the middle of July, when it had thousands of flowers upon
it, which, with thousands that must still succeed, might very probably produce
more than 50000 seeds more, even supposing many of the flowers to produce
no seed, considering 1000 rings contain 12000 seeds and more; and if we
multiply the number of rings actually counted, by 14, the number of seeds
contained in one ring, instead of 12, we shall have an addition of 20O0O
seeds, all which, added together, amount to 200000, the possible increase of
one seed.

On the Nature of Amber. By John Ambrose Bearer. N° 468, p. 322.

From the Latin.

M. Beurer cannot admit that amber is the resinous juice of a tree, for these
reasons: Is it probable that amber should pass out of the earth into the sea ? or
whence is that passage ? since the trees are not near the sea.

Again, can this resin pass through the earth like water, or difi^use itself so
copiously through it ? If this were possible, would it not rather grow stift', and
adhere to the surface of the earth.

Besides, the heat of the sun can never cause such a flood of resins, as to fill
several subterraneous tracts. For resins exude by drops, the least part of which



632 PHILOSOPHICAl. TRANSACTIONS. [aNNO 1742-3.

only reach the ground, the greater part adhering to the bark of the tree. Be-
sides, amber is often found on mountains, and in pits, where trees were never
planted. And as to the arguments from the distillation of vitriolic acid with
turpentine, they do not prove the case ; for though something bituminous is
then produced, it is not real amber, as it wants the equal mixture, transparence,
elasticity, and hardness. This may be easily and quickly produced, by the
mixture of any distilled ethereal oil, coagulated with vitriolic acid ; from
which mixture there presently arises something bituminous, but not amber.

Amber probably derives its origin from a mineral, viz. from a soft bitumen
(oleum naphtae) and a sulphureous vitriolic acid, which mix in the form of steam,
and presently harden. — This is also proved by the fossil amber; for wherever
this is dug up, there are also found, among the' blue clay, bituminous wood,
coal, vitriol, and often alum. And the amber found in the sea, is produced in
the same manner as that formed in the mountains, being only washed out of the
earth by the beating of the waves, and partly lost in the deep, partly thrown up
on the banks.

Further, as the vitriolic acid, with the bitumen, produces the form and
semblance of amber; that acid will quite dissolve it again, and leave it in the
same state, without destroying any constituent part, reducing its hardness,
transparence, and elasticity.

An Account* of a Booh intitled, A Treatise of Fluxions, in Two Books. By
Colin Maclaurin, A. M., F. R. S. 2 Fols. Ato. N" 468, p. 325.

The author's first design, in composing this treatise, was to establish the
method of fluxions on principles equally evident and unexceptionable with those
of the ancient geometricians, by demonstrations deduced after their manner, in
the most rigid form, and by illustrating the more abstruse parts of the doctrine,
to vindicate it from the imputation of uncertainty or obscurity. But he has
likewise comprehended in this work the application of fluxions to the most im-
portant geometrical and philosophical inquiries. It consists of an introduction,
and two books. In the introduction he gives an abstract of the discoveries of
the ancients in the higher parts of geometry, with observations on their method,
and those that first succeeded to it. The first book treats of fluxions in a geo-
metrical method, and the second treats of the computations.

In the introduction we have an abstract not only of the discoveries of the
ancients in the higher parts of geometry, but likewise of their demonstrations.
After an account of the propositions of this kind, that are to be found in the

• This very able and masterly account was probably the composition of the excellent author
himself.



VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 633

twelfth book of Euclid, there follows a summary of what is most material in the
treatises of Archimedes, concerning the sphere and cylinder, conoids and
spheroids, the quadrature of the parabola and the spiral lines. The demon-
strations are not precisely in the same form as those of Archimedes, but are
often illustrated from the elementary propositions concerning the cone, or
corollaries from them, after the example of Pappus, (Coll. Math. Prop. 21st,
lib, 4) from whom a proposition is demonstrated, and rendered more general,
concerning the area of the spiral generated on a spherical surface by the com-
position of two uniform motions, analogous to those by which the spiral of
Archimedes is described on a plane. This area, though a portion of a curve
surface, is found to admit of a perfect quadrature, and this proposition con-
cludes the abstract. He takes occasion from these theorems to demonstrate
some properties of the conic sections, that are not mentioned by the writers on
that subject; and there are more of this kind described in the llth and 14th
chapters of the first book.

It is known, that if a parallelogram, circumscribed about a given ellipse, have
its sides parallel to the conjugate diameters, then shall its area be of an invaria-
ble or given magnitude, and equal to the rectangle contained by the axes of the
figure ; but this is only a case of a more general proposition. For if, upon any
diameter produced without the ellipse, you take two points, one on each side of
the centre at equal distances from it, and the four tangents be drawn from these
points to the ellipse, those tangents shall form a parallelogram, which is always
of a given or invariable magnitude, when the ellipse is given, if the ratio of
those distances to the diameter be given ; and when the ratio of those distances
to the semidiameter is that of the diagonal of a square to the side, (or of
n/l to l) the parallelogram has its sides parallel to conjugate diameters. It is
likewise shown here, how the triangles, trapezia, or polygons of any kind are
determined, which, circumscribed about a given ellipse, are always of a given
mag



nit



itude.
There is also a general theorem concerning the frustum of a sphere, cone,
spheroid, or conoid, terminated by parallel planes, when compared with a
cylinder of the same altitude on a base equal to the middle section of the
frustum made by a parallel plane. The difference between the frustum and the
cylinder, is always the same in different parts of the same, or of similar solids,
when the inclination of the planes to the axis, and the altitude of the frustum,
are given. This difference vanishes in the parabolic conoid. It is the same in
all spheres ; being equal to half the content of a sphere of a diameter equal to
the altitude of the frustum. In the cone it is -j-th of the content of a similar

VOL. VIII. 4 M



634 PHILOSOPHICAL TRANSACTIOKS. [aNNO 1742-3.

cone of the same height with the frustum ; and in other figures it is reduced to
the difference in the cone.

In the remarks on the method of the ancients, the author observes, that they
established the higher parts of their geometry on the same principles as the ele-
ments of the science, by demonstrations of the same kind ; that they seem to
have been careful not to suppose any thing to be done, till by a previous pro-
blem they had shown how it was to be performed : far less did they suppose any
thing to be done, that cannot be conceived to be possible, as a line or series to
be actually continued to infinity, or a magnitude to be diminished till it be-
comes infinitely less than it was. The elements into which they resolved mag-
nitudes were always finite, and such as might be conceived to be real. Un-
bounded liberties have been introduced of late, by which geometry, wherein
every thing ought to be clear, is filled with mysteries, and philosophy is like-
wise perplexed. Several instances of this kind are mentioned. The series 1,
2, 3, 4, 5, 6, 7, &c. is supposed by some to be actually continued to infinity ;
and, after such a supposition, we are puzzled with the question, whether the
number of finite terms in such a series is finite or infinite. In order to avoid
such suppositions, and their consequences, the author chose to follow the anci-
ents in their method of demonstration as much as possible. Geometry has been
always considered as our surest bulwark against the subtleties of the sceptics,
who are ready to make use of any advantages that may be given them against
it ;* and it is important, not only that the conclusions in geometry be true, but
likewise that their evidence be unexceptionable. However, he is far from
affirming, that the method of infinitesimals is without foundation, and after-
wards endeavours to justify a proper application of it.

The grounds of the method of fluxions are described in chap. 1, book ], and
again in chap. 1, book 2. In the former, magnitudes are conceived to be
generated by motion, and the velocity of the generating motion is the fluxion
of the magnitude. Lines are supposed to be generated by the motion of points.
The velocity of the point that describes the line is its fluxion, and measures the
rate of its increase or decrease. Other magnitudes may be represented by lines
that increase or decrease in the same proportion with them ; and their fluxions
will be in the same proportion as the fluxions of those lines, or the velocities of
the points that describe them. When the motion of a point is uniform, its
velocity is constant, and is measured by the space which is described by it in a
given time. When the motion varies, the velocity at any term of the time is
measured by the space which would be described in a given time, if the motion

• See Bayle's Dictionary, Artide Zeno. — Orig.



TOli. XLII.] PHILOSOPHICAL TRANSACTIONS. 635

was to be continued uniformly from that term without any variation. In order
to determine that space, and consequently the velocity which is measured by it,
four axioms are proposed concerning variable motions, two concerning motions
that are accelerated, and two concerning such as are retarded. The first is,
that the space described by an accelerated motion is greater than the space
which would have been described in the same time, if it had not been accelerated,
but had continued uniform from the beginning of the time. The second is,
that the space which is described by an accelerated motion, is less than the
space which is described in an equal time by the motion which is acquired by
that acceleration continued afterwards uniformly. By these, and two similar
axioms concerning retarded motions, the theory of motion is rendered applica-
ble to this doctrine with the greatest evidence, without supposing quantities in-
finitely little, or having recourse to prime or ultimate ratios. The author first
demonstrates from them all the general theorems concerning motion, that are
of use in this doctrine ; as, that when the spaces described by two variable mo-
tions are always equal, or in a given ratio, the velocities are always equal, or in
the same given ratio ; and conversely, when the velocities of two motions are
always equal to each other, or in a given ratio, the spaces described by those
motions in the same time are always equal, or in that given ratio ; that when a
space is always equal to the sum or difference of the spaces described by two
other motions, the velocity of the first motion is always equal to the sun) or
difference of the velocities of the other motions ; and conversely, that when a
velocity is always equal to the sum or difference of two other velocities, the
space described by the first motion is always equal to the sum or difference of
the spaces described by these two other motions. In comparing motions in this
doctrine, it is convenient and usual to suppose one of them uniform ; and it is
here demonstrated, that if the relation of the quantities be always determined
by the same rule or equation, the ratio of the motions is determined in the
same manner, when both are supposed variable. These propositions are de-
monstrated strictly by the same method which is carried on in the ensuing
chapters for determining the fluxions of the figures.

In chap. 1, a triangle that has two of its sides given in position, is supposed
to be generated by an ordinate moving parallel to itself along the base. When
the base increases uniformly, the triangle increases with an accelerated motion,
because its successive increments are trapezia, that continually increase. There-
fore, if the motion with which the triangle flows, was continued uniformly from
any term for a given time, a less space would be described by it than the incre-
ment of the triangle which is actually generated in that time by axiom 1, but u
greater space than the incremeiit which was actually generated in an equal time

4M2



636 PHILOSOPHICAL TRANSACTIONS. [aNNO 1742-3,

preceding that term, by axiom 2, and hence it is demonstrated, that the fluxion
of the triangle is accurately measured by the rectangle contained by the corres-
ponding ordinate of the triangle, and the right line which measures the fluxion
of the base. The increment which the triangle acquires in any time, is re-
solved into two parts ; that which is generated in consequence of the motion
with which the triangle flows at the beginning of the time, and that which is
generated in consequence of the acceleration of this motion for the same time.
The latter is justly neglected in measuring that motion, or the fluxion of the
triangle at that term, but may serve for measuring its acceleration, of the 2d
fluxion of the triangle. The motion with which the triangle flows, is similar
to that of a body descending in free spaces by a uniform gravity, the velocity of
which, at any term of the time, is not to be measured by the space described
by the body in a given time, either before or after that term, because the mo-
tion continually increases, but by a mean between these spaces.

When the sides of a rectangle increase or decrease with uniform motions, it
may be always considered as the sum or difference of a triangle and trapezium ;
and its fluxion is derived from the last proposition. If the sides increase with
uniform motions, the rectangle increases with an accelerated motion ; and in
measuring this motion at any term of the time, a part of the increment of the
rectangle, that is here determined, is rejected, as generated in consequence of
the acceleration of that motion.

The fluxions of a curvilineal area (whether it be generated by an ordinate
moving parallel to itself, or by a ray revolving about a given centre) and of the
solid, generated by the area revolving about the base, are determined by de-
monstrations of the same kind; and when the ordinates of the figure increase,
the increment of the area is resolved in like manner into two parts, one of
which is only to be retained in measuring the fluxion of the area, the other be-
ing rejected as generated in consequence of the acceleration of the motion with
which the figure flows. An illustration of 2d and 3d fluxions is given by re-
solving the increment of a pyramid or cone into the several respective parts that
are conceived to be generated in consequence of the 1st, 2d, and 3d fluxions
of the solid, when the axis is supposed to flow uniformly.

In chap. 5, a series of lines in geometrical progression are represented by an
easy construction. The first term being supposed invariable, and the second to
increase uniformly, all the subsequent terms increase with accelerated motions.
The velocities of the points that describe those lines being compared, it is de-
monstrated, from the axioms by common geometry, that the fluxions of any
two terms are in a ratio compounded of the ratio of the terms, and of the ratio
of the numbers that express how many terms precede them in the progression.



VOL. XLII.] PHILOSOPHICAL TKANSACTIONS. 637

In the 6th chapter, the nature and properties of logarithms are described
after the celebrated inventor ; and it is observed, that he made use of the very
terms fluxiis and fluat on this occasion. A line is said to increase or decrease
proportionally, when the velocity of the point, that describes it. is always as
its distance from a certain term of the line ; and if in the mean time another
point describes a line with a certain uniform motion, the space described by the
latter point is always the logarithm of the distance of the former from the given
term. Hence the fluxion of this distance is to the fluxion of its logarithm, as
that distance is to an invariable line ; and the fluxions of the quantities that
have their logarithms in an invariable ratio, are to each other in a ratio com-
pounded of this invariable ratio, and of the ratio of the quantities themselves.
Some propositions are demonstrated, that relate to the computation of lo-
garithms ; but this subject is prosecuted further in the second book. The lo-
garithmic curve is here described, with the analogy between logarithms and
hyperbolic ratios.

In the 7 th chapter, after a general definition of tangents, it is demonstrated,
that the fluxions of the base, ordinate, and curve, are in the same proportion
to each other, as the sides of a triangle respectively parallel to the base, ordi-
nate, and tangent. When the base is supposed to flow uniformly, if the curve
be convex towards the base, the ordinate and curve increase with accelerated
motions ; but their fluxions at any term are the same as if the point which de-
scribes the curve had proceeded uniformly from that term in the tangent there.
Any further increment which the ordinate or curve acquires, is to be imputed
to the acceleration of tlie motions with which they flow. A ray that revolves
about a given centre, being supposed to meet any curve and an arc of a circle
described from the same centre, the fluxions of the ray, curve, and circular
arc, are compared together ; and several other propositions concerning tangents
are demonstrated from the axioms. The next chapter treats of the fluxions of
curve surfaces in a similar manner.

The Qth chapter treats chiefly of the greatest and least ordinates of figures,
and of the points of contrary flexure and cuspids. The fluxion of the base
being given, when the fluxion of the ordinate vanishes, the tangent becomes
parallel to the base, and the ordinate most commonly is a maximum or mini-
mum, according to the rule given by authors on this subject. But if the second
fluxion of the ordinate vanish at the same time, and the third fluxion be real,



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 73 of 85)