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itself, on the uses of this sort of quadrature by the limiting polygons, because
it is one of that kind which requires no other knowledge but what depends on
the common properties of number and magnitude; and so may serve as an
instance to show, that no other is requisite for the establishment of principles


for arithmetic and geometry. A truth which, though certain in itself, may
perhaps seem doubtful from the nature and tendency of the present inquiries in
mathematics. For, among the moderns, some have thought it necessary, for
investigating the relations of quantities, to have recourse to very hard hypo-
theses ; such as that of number infinite and indeterminate ; and that of mag-
nitudes in statu fieri, existing in a potential manner, which are actually of no
magnitude. And others, whose names are truly to be reverenced on account
of their great and singular inventions, have thought it requisite to have recourse
even to principles foreign to mathematics, and have introduced the considera-
tion of efficient causes, and physical powers, for the production of mathemati-
cal quantities ; and have spoken of them, and used them, as if they were a
species of quantities by themselves.

N. B. In the following proposition Mr. Machin has, for the sake of brevity,
made use of a peculiar notation for composite numbers, or such quantities as
are analogous to them, whose factors are in arithmetical progression.

The quantity expressed by this notation has a double index: that at the head
of the root at the right-hand, but separated by a hook to distinguish it from the
common index, denotes the number of factors; and that above, within the
hook on the left hand, denotes the common difl^erence of the factors proceed-
ing, in a decreasing or increasing arithmetical progession.

Thus the quantity — ^(""denotes, by its index m on the right hand, that

it is a composite quantity, consisting of so many factors as there are units in
the number m; and the index a above, on the left, denotes the common differ-
ence of the factors, decreasing in an arithmetical progression, if it be positive;
or increasing, if it be negative; and so signifies, in the common notation,
the composite number or quantity, n -{■ a.n -\- a — a. n-f-a — 2a.. n-^ a — 3a.
and so on.


For example : — ^^if" is = 7z-j-5. n-f-3. n-|- l.n — \,n — 3. n — 5. con-
» + 5

sisting of six factors whose common difference is 2. After the same manner

^(^ is = n -|- 4. n -|- 2. n. n — 2. n — 4, consisting of five factors. Accord-
it + 4

ing to which method it will easily appear, that if a be any integer, then
j=l{^'' + ^ will be = nn — 1. nn — Q nn — 25, continued to such a num-

n + 2a + X

bar of double factors, as are expressed by a -|- 1, or half the index, which in


this case is an even number. So ^('•' + ' will be equal to

K + 2a

n.nn — 4.nn — \6. nn — 36, and so on, where there are to be as many double


factors as with one single one (n) will make up the index 2a -{■ I, which is an
odd number.

If the common difference a be an unit, it is omitted:
Thus, ji^" is = n. 7J — ].n — 2.n — 3. n — 4.n — 5, containing six factors.
So of is = 6. 5. 4. 3. 2. 1 , and the like for others.

If the common difference a be nothing, then the hook is omitted, and it


becomes the same with the geometrical power: so — i:^ ('" is = 71 _|_ I" according

to the common notation.

Proposition 1. — ^n arch /ess than a semicircle being given, with a point in
the diameter passing through one of its extremities; to Jind by means of the sine
of a given part of the arch less than one half, the area of the sector subtended
by the given arch, and comprehended in the angle made at the given point. — Let
PNA, fig. 7. p'- 6, be a semicircle described on the centre c, and diameter ap,
and let pn be the given arch less than a semicircle, and s the given point in the
diameter ap, passing through one extremity of the arch np in p. Then taking
any number n greater than 1, let pk be an arch in proportion to the given arch
PN, as unity to the number n; and let it be required to find, by means of the
sine of the arch pk, the area of the sector nsp, subtended by the given arch
NP, and comprehended in the angle nsp made at the given point s.

From N and k let fall, on the diameter ap, the perpendiculars nm and kl,
and join cn and ck. Then let t stand for cp, the semidiameter of the circle;
yfor cs, the distance of the given point s from the centre; p for sp, the dis-
tance of it from the extremity of the arch, through which the diameter ap
passes; and y for kl, the sine of the arch kp in the given circle.

These substitutions being presupposed, the problem is to be divided into two
cases; one when sp is less, and the other when it is greater than the semidia-
meter CP.

Case 1 . — If SP be less than cp, then take an area h equal to the sum of the

rectangles expressed by the several terms of the following series continued ad

libitum :

2 2 2

^l' __-^|4 '^ '»

PS _I_ < + "+ ''x/ ^ f , 9<-n + 3lx/ ^ y' , 9_X_25M- n + 5lxf ^ ^ , -
1 3j3 t 55 t J.7 t^

And the area -^n X h will determine the area of the sector nsp ad libitum.

For the sector psn, being the excess of the sector ncp above the triangle
ncs, will be the difference of two rectangles : -J-cp X pn — 4-cs X nm; but pn
is the multiple of the arch pk, namely n X pk ; and nm is the sine of that


multiple arch ; therefore if for cp be put t, for cs, /, according to the suppo-
sition; and if for pk be substituted ^-+4r X 4' + lr X^l + t^m x^ + &c.

by cor. 2, lem. 2; and for nm

,- ^ X 75 H i7- X r* ^=^ X -^^ + &c. according to cor. 1,

1 3|, t 5,5 t 717 t ^

lem. 1, the area of the sector will appear in a series, as is above determined.

But since the number n is greater than 1, and the given arch pn is less than
a semicircle, and consequently kl or ?/, the sine of the submultiple arch pk, is
less than the semidiamer cp or t\ it may thence be easily proved, that the series
will approximate to the just quantity of the area, ad libitum.

Corol. 1. — Hence, if the number n be taken equal to \/5 ^.'^25 + ^> the

sector Nsp will be = \npy + ^—^^"7^-^/ + **** + J^.y'' + &c.

For the numerator of the coefficient of the third term in the series, that

determines the area h, namely, 9; — n+~3| x f, 'S equal to 9^ — nn— I .nn — Q.f,
which, according to the above determination of the number n, will become
nothing; therefore, if for < — p be puty in the second term, and the value of
n be substituted for n in the third and fourth, the series for the area will appear
on reduction to be as is here laid down.

Corol. 2. — Hence the area of the sector nsp may be always defined nearly by
the terms of a cubic equation.

For the number n, as constructed in the former corollary, is always greater
than the square root of 10, and consequently ^ is always less than the sine of

one-third part of the given arch ; so that the fourth term „ y, with the sum

of all the following terms of the series, can never be more than a small part of

the whole sector.

Corol. 3. — If K stand for 57,2957795 &c. degrees, or the number of degrees

contained in an angle subtended by an arch of the same length with the radius

of the circle, and m be the number of degrees in an angle which is to four

right angles, as the area nsp to the area of the whole circle; then will m be

= ^1X^ + nH-n:r^^LhI X %, nearly.

For - X - will appear, by the construction, to be equal to the sector nsp.
Case 1. — If sp be greater than cp, then take an area h equal to the sum of
the terms in the following series:


2 2 2


SI + ^-"H- lIx/ y^t ^ 9^+» + 3'x/ X ^ + 9X 25/ -« + 51x7 X y _j. gjj._
1 ^ <* 5I <* tTj '" '

and the area -J-m X h, will be the sector, as before.

For the point s being on the contrary side of the centre, to what it was be-
fore, it will easily appear, that the change of -\-f into — f, must reduce one
case to the other, without any other proof.

Carol. — Hence, if the number n be taken equal to ^Z —r-, or in this case y'^,
then the series for the sector will want the second term, as in the former it
wanted the third.

Definition. — The angle called by Kepler the anomalia eccentri, is a fictitious
angle in the elliptic orbit of a planet, being analogous to the area described by
a line from the centre of the orbit, and revolving with the planet from the line
of apsides; in like manner as the mean anomaly is a fictitious angle, analogous
to the area described by a line from the focus.

Otherwise, if c be the centre, s the focus of an elliptic orbit described on
the transverse axis ap, and the area nsp in the circle be taken in proportion to
the whole, as the area described in the ellipsis about the focus, to the whole:
then is the arch of the circle pn, or the angle nop, that which Kepler calls the
anomalia eccehtri.

This angle may be measured either from the aphelion, or from the perihe-
lion: in the following proposition it is supposed to be taken from the peri-

Proposition 1. — The mean anomaly of a comet or planet, revolving in a given
elliptic orbit, being given; to find the anomalia eccentri. — The solution of this
problem requires two different rules; the first and principal one serves to make
a beginning for a further approximation, and the other is for the progression
in approximating nearer and nearer ad libitum.

1. The rule for the first assumption : let ^,/, and j&, stand, as before, for the
semi-transverse axis of the ellipsis, the semi-distance of the foci, and the peri-
helion distance ; then taking the number n equal to \/ 5 + ^25 + ^j let t stand

for ■ — ; and p for—- — ; — , or^T; which constant numbers, be-

nnt — nn — l.p nnt — nn—\.p t '

ing once computed for the given orbit, will Serve to find the angle required
nearly by the following rule.

Let M be the number cf degrees in the angle of mean anomaly to the given
time, reckoned from or to the perihelion ; and supposing r, as before, to stand
for 57,2937 &c. degrees; take the number n = y'— m, and let A be the angle

VOL. VIIT. 1 atlJ o B B


whose sine is n V ^+ \/j + $ + n V g "" V i + '^'' *^^^" ^^^ multiple
angle n X A, will be nearly equal to the anomalia eccentri.

The truth of which will appear from the resolution of the cubic equation, in
the last corollary to the preceding proposition.

Carol. ] . — If the quadruple of the quantity — ^ be many times greater or many
times less than unity; or, which amounts to the same, if the mean anomaly n,
be many times less, or many times greater, than the angle denoted by the given
quantity -^Ry^p, one or the other of which two cases most frequently happens
in orbits of very large eccentricity; then the theorem will be reduced to a sim-
pler form, near enough for use.

Case 1 . — If M be many times less than
^ Rv^p, then the angle a may be taken for that whose sine Is —.

Case 2.— If M be many times greater than
— R\/p, then let a be the angle whose sine is n — -; and the multiple angle
n X A, according to its case, will be nearly equal to the angle required. '

Carol. 1. — In orbits of very large eccentricity, the perihelion distance p is
many times less than the semi-distance of the foci /, and the number

n = \/5+ ^25 + ^; is always nearly equal to '/lO, or to the integer 3, either

of which may be used for it, without any material error in the orbits of comets.

2. The rule for a further correction ad libitum.

Let M be the given mean anomaly, t the semi-transverse axis, as before; and
let B be equal to, or nearly equal to, the multiple angle n X A, before found;
then if p. be the mean anomaly, and x the planet's distance from the sun, com-
puted to the anomalia eccentri b ; the angle b taken equal to b -| — X m — /*,
will approach nearer to the true value of the angle sought; and by repetitions
of the same operation, the approximation may be carried on nearer and nearer,
ad libitum.

This last rule being obvious, the explication of it may be omitted at present.

Scholium. — In this solution, where the motion is reckoned from the peri-
helion, the rule is universal, and under no limitation. But had the motion
been taken from the aphelion, the problem must have been divided into two
cases: one is, when the eccentricity is less than -^x the other is, when it is
not less, but is either equal to, or more than in that proportion.

If the eccentricity be not less than -rV> then the same rule will hold, as be-
fore, only putting the aphelion distance, suppose a instead of the perihelion
distance p, and substituting — / for -^f in the rule for the number n.


If the eccentricity be less than .V, then take the number n equal to
v/^ and — X - will be nearly equal to the sine of the submultiple part of the
anomalia eccentri, denominated by the number n, as before.

It is needless to observe, that the like rules would obtain in hyperbolic orbits,
mutatis mutandis. But that which perhaps may not appear unworthy of being
remarked, concerning this sort of solution from the cubic root, is, that
though the rule be altogether impossible, on a total change of the figure of
the orbit, either into a circle, or into a parabola; yet it will operate so much
better, and stand in need of less correction, according as the figure advances
nearer, in its change, towards either of those two forms.

That the use of the method may better appear, it may not be amiss to add a
few examples.

The following are two for the orbits of planets, one the most, and the other
the least eccentric; but which are more to show the extent of the rule, than to
recommend the use of it in such cases; for there are many other much better,
and more expeditious methods, in orbits of small eccentricity. The other two
examples are adapted to the orbits of two comets, whose periods have been
already discovered by Dr. Halley; the one is to show the use of one of the
rules in the first corollary, and the other is to explain the use of the other rule.

Example 1 . — For the Orbit of Mercury. — If an unit be put for the semi-
transverse axis t, the eccentricity 0,20589 ^'^^ become f, and the perihelion
distance J!) will be 0,79^11 ; therefore by means of the number b, given as be-
fore, the constant numbers for this orbit will appear to be, n = 3,56755,
T = 0,5857271, p = Y T = 0,4651319, and hence — 5Z_ = 0,0085965.

Example. — Suppose m, the mean anomaly from the perihelion, to be 120"
00' 00", to which it is required to find the anomalia eccentri.

Here, since the mean anomaly m is not many times more than the limiting
angle -^R-v/p, which in this orbit is about 74°, recourse must be had to the

general rule in the proposition.

The number n then, which is (/— m, will be = 1,0104195 ; which, found,

gives nV i -f \/i + ^ = 1,0389090; and also N\/i - ^/i + -^= -
0,4477126. Therefore the sum of both, under their proper signs, viz.
0,591 1964, will be the sine whose arch 36°,24195 is the angle A; the multiple
of which n X A = 129°,295503, will be the angle to be first assumed for the
anomalia eccentri.

For a further correction ; this angle, now called b, whose sine is suppose y,

BB 2


and its cosine z, gives, by a known rule, f + ^z = 1,1304, for x the planet's

distance from the sun ; and by another known rule b — —y = 1 20°, 1 6568, for u.
the mean anomaly to the anomalia eccentri b. Therefore the correct angle
B, = B + ^ X M — |«, will be 129°,14846 = 129°8'54",5, erring, as will ap-
pear from a further correction, about -rV of a second.

This angle, being thus determined, will give by the common methods,
137" 48' 33-t", for the true anomaly, or angle at the sun : the sine of the true
anomaly being in proportion to the sine of the anomalia eccentri, as the semi-
conjugate axis, to the planet's distance from the sun. So that the equation of
the centre in this example is 17° 48' 33^".

Example 2. — For the Orbit of Venus. — Supposing, as before, the mean
distance t to be unity, and the eccentricity/ to be 0,0069855 ; the constant
numbers for this orbit will be, p = 0,99301 15 ; n — 6,41 16 ; t = 1,562134 ;

I' = 0,1551217 ; ^ = 0,0127571 ; and the limiting angle, ^r\/t-, will ap-
pear to be about 303 degrees.

Example. — Let m be 120° OO' OO", as in the former example. Then, since
the mean anomaly is, in this case, not many times less than the limiting angle,
the general rule must be used as before ; according to which the number n will
appear to be 1,152585; the sine of a will be 0,32179)7; the angle a,
18°,77I32; and the multiple re X A, or angle b, for the first assumption of
the anomalia eccentri, will be 120°,354l6.

This angle b will give, by the method before explained, the angle b =
120°,34555, or 120° 2l' 44* fer^, for the anomalia eccentri correct ; the error
of which will appear, on examination, to be but a small part of a second.

In this example, the true anomaly is 120° 4l' 25", 1 ; and consequently the
equation of the centre no more than 41' 25", 1.

Example 3. — For the Orbit of the Comet of l682. — To know the mean
anomaly of this comet, to any given time, it is to be premised, that it was at
the perihelion in the year l682, on Sept 4, at 21'' 22"*, equated time to the
meridian of Greenwich, and makes its revolution about the sun, as Dr. Halley
has discovered, in 754- years.

The perihelion distance p is, according to his determination, 0,0326085 parts
of the mean distance t. So that the constant numbers for the orbit will be,
n = 3,1676061 ; T = 0,2054272; p = 0,00669867 ; and the Hmiting angle,
-^R\/p, will be about I9', or -f of a degree.


In the orbits of comets, the rule for the first assumption of the anomalia
eccentri, is generally sufficient without correction.


ThuSj suppose the mean anomaly m to be 0,072706, (as it was at the time of
an observation made at Greenwich on August 30, \682, at 7^ 42"" eq. t,) then
the general rule (which must be here used, since the angle of mean anomaly is
not above 4 or 5 times less than the limiting angle) will give n X a or b = 2°
12' 48", 7, erring about ^^ of a second from the true anomalia eccentri.

But in these orbits, the rules in the first corollary to the second proposition
most frequently take place, especially the last ; and the calculation may also be
further abbreviated, by putting the square root of 10, or the integer 3, for the
number n.

Example. — Suppose the mean anomaly to be 0°,006522, or 23",4792 : here,
since m is 50 times less than the limiting angle, the rule in the first case of
the first corollary may be used ; that is, to take the sine of the angle a =

< X M

np X R

Therefore, if the number 3 be put for n, the sine of a, which is —.will be

= 0,001 16367 ; and consequently the angle A will be 4'0",011 ; and the mul-
tiple angle n X a, to be assumed for the anomalia eccentri, will be 12'0",033,
the error of which will be found to be about ^ of a second.

Example 4. — For the Orbit of the great Comet of the Year l680. — ^This
comet, according to Dr. Halley, performs its period in 575 years ; and was in
its perihelion on Dec. 7, 168O, at 23*' OQ"* eq. t. at London ; the perihelion
distance p is 0,000089301, in parts of the mean distance t: therefore sup-
posing the number n to be v'lO, the constant numbers for the orbit will be
T = 0,2000161 ; p = 0,000017862, and the limiting angle ^r^p, will be
about -^ of a second.

Example. — Suppose the mean anomaly to be 3'31*,4478, or 0°,0587354],
(as it was at the time of the first observation made on it in Saxony, on Nov. 3,
at 16** 47™ eq. t, at London,) here, since the mean anomaly is many times
greater than -^ of a second, the rule in the second case of the first corollary may
be used ; that is, by taking the sine of a = n .

3t p

But the number n or^ — m is = 0,05794134 ; and - will be = 0,0030827;


therefore (n — - =) 0,0576330/, will be the sine whose arch 3'',30397 is the
angle a; and the multiple angle n X a = 10° 26' 53*,05, will be the angle to
be first assumed for the anomalia eccentri ; the error of which will be found to
be less than a second.

The true anomaly, computed from this angle according to the rule in the
example for Mercury, will appear to be 171° 38' 24", from the perihelion.


By these examples it appears, that the solution is universal in all respects; for
the first two, compared with the last two, serve to show that it is not confined
to any particular parts of the orbit, but extends to all degrees of mean anon)aly:
and by comparing the second with the last, it sufficiently appears to be univer-
sal with respect to the several degrees of eccentricity ; since in one the equation
of the centre, for the reduction of the mean to the true motion, is not so
much as the 1 70th part of the whole ; whereas in the other it amounts to almost
3000 times as much as the mean motion itself.

Postscript. — On reviewing the reflections on the quadrature of the circle,
Mr. M. believes it may be necessary, to prevent any mistake that may arise
from the different opinions that obtain about the nature of mathematical quan-
tity, to explain himself a little on that head ; as also to add a few words to show
how the method of quadrature, by limiting polygons, takes place in other
figures, as well as the circle.

He takes then a mathematical quantity, and that for which any symbol is put,
to be nothing else but number with regard to some measure which is considered
as one. For we cannot know precisely and determinately, that is, mathemati-
cally, how much any thing is, but by means of number. The notion of con-
tinued quantity, without regard to any measure, is indistinct and confused ; and
though some species of such quantity, considered physically, may be described
by motion, as lines by points, and surfaces by lines, and so on ; yet the mag-
nitudes or mathematical quantities are not made by that motion, but by
numbering according to a measure.

Accordingly, all the several notations that are found necessary to express the
formations of quantities, refer to some office or property of number or measure ;
but none can be interpreted to signify continued quantity, as such.

Thus some notations are found requisite to express number in its ordinal
capacity, or the numerus numerans, as when one follows or precedes another,
in the first, second or third place, from that on which it depends ; as the quan-
tities X, r, X, oc, X, referring to the principal one x.

So, in many cases, a notation is found necessary to be given to a measure, as
a measure ; as for instance. Sir Isaac Newton's symbol for fluxion x ; for this
stands for a measure of some kind, and accordingly he usually puts an unit for
it, if it be the principal one on which the rest depend.

So some notations are expressly to show a number in the form of its com-
position, as the index to the geometrical power x", denoting the number of
equal factors which go to its composition, or what is analogous to such.

But that there is no symbol or notation, but what refers to discrete quantity,
is manifest from the operations, which are all arithmetical.


And hence it is, there are so many species of mathematical quantity, as there
are forms of composite numbers, or ways in their composition ; among which
there are two, more eminent for their simplicity and universality, than the rest:
one is the geometrical power formed from a constant root ; and the other,
though well known, yet wanting a name as well as a notation, may be called the
arithmetical power; or the power of a root uniformly increasing or diminishing;
the one is only for the form of the quantity itself, the other is for the constitu-
tion of it from its elements.

Now from the properties of either of these, it would be easy to show how
the quadratures of simple figures are deducible from the areas of their limiting
polygons. Mr. M. just points out the method from the arithmetical power, as
being the shortest and readiest at hand.

Let z, 2, z, &c. or z, z, z, &c be quantities in arithmetical progression,

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 23 of 85)