Royal Society (Great Britain).

# 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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Font size that came out before ; and thus it will revert to the same in every case of radi-
cality whatever.

To try therefore whether the expression "V a -\- Â»/ b can be reduced to a sim-
pler state; put 2x = ^/ a + v^i -f 1/ a â€” Â»/ b, and ^aa â€” Z; = m, also y =
XX â€” m; then the expression reduced will be a? + ^y, if it can admit of in-
tegral, or at least rational quantities.

But in case these should not be integer or rational quantities, yet the rule
above will be of use in solving equations of any kind, as will be seen here-
after.

In the mean time perhaps this doubt may arise, whether this rule will obtain
universally in any powers whatever of the binomial, viz. whether in any bino-
mial, whose index is n, if from the square of the sum of the terms in the un-
even places, there be subtracted the square of the sum of those in the even
places, the remainder will be another binomial having the index n. To which
Mr. Demoivre answers, that this is a fact which has been observed by many

VOL. VIII. N N

274 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

writers, and therefore may be considered as confirmed by experience. But as
he has never seen any demonstration of it, he adds the following.

Take the binomial (x -\- y), and expand it ; take also this other binomial
{x â€” yY, which also expand ; put {x + y)" = i, and {x â€” y)n = p â–  now it
is evident that, if the expanded binomials be united by addition, their sum will
give double the sum of the uneven terms of the first binomial ; but if the latter
be subtracted from the former, then the remainder will be double the sum of
the even terms of the same binomial ; hence it follows that â€”â€” is the sum of

the uneven terms, and ^-^ the sum of the even terms.

From "i+lPl+JP^ the square of the first sum, taken '"^P^ + PP ^ the
square of the last, the remainder will be ps = (x -\- y)' X {x â€” y)" =
(xx â€” yy)", the nth root of which is xx â€” yy.

Carol. â€” By putting 1x =â–  V a â– \- ^/b + V a â€” V b, and making Vaa â€” h
= TO, then expounding n successively by 1,2, 3, 4, 5, 6, &c. there will arise
the following equations :

1st.

X =

a

2d.

Ix" â€”

m =

a

3d.

Ax^ -

3mx =

a

4th.

Sx* -

Bmx'' +

m* =

a

5th.

\6x' -

20ma^ +

dm^x =

a

6th.

six" â€”

48TOX* +

ISmV â€”

m^ =

7th. OAx' â€” li2mx^ -f 56toV â€” 7m^x = a
&c.

Now these equations are of the same form as the equations for cosines,
though they are things of a quite different nature. Thus, let r be the radius of
a circle, c the cosine of any given arc, and x the cosine of another arc, which
is to the former, as 1 to n. Then it will be,

1st. X =: c

2d. 20^ â€” r^ = re

3d. Ao^ â€” 3r^x = r'^c

4th. Bx* â€” BrV 4- r* = r^c

5th. iQx'' â€” lOr'a? + 5r*x =. r*c

6th. 32a?* â€” 48rV + 18rV â€” r" = r^c

7th. QAkP â€” 112rV + 56rV â€” Tr^x â€” r^c

&c.

VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 275

Now putting r = 1, for brevity, then the general form of these is
2- Xaf- 2-' X ^x- + 2- X J . ^^-^ - 2-' X Â» !i=l. Izf^-V

1 12 X 2 3

&c. ^ c.

The difference in these equations consi sts chiefl y in this , (hat the former are
derived from the equation 2x = 1/ a + v'^ + "V a â€” ^h, but the latter from
the equation 1x =:. 1/ a -\- */ â€” b â€” V a â€” n/ â€” b; and if this latter equation
be freed from its general radicality, there will be obtained equations for the
cosines.

Let therefore the equation 1x =. \/a-\-'s/â€” b-\-\/a â€” Â»/~b be pro-
posed to be freed from its radical sign ]/ .

Put \/ a + ^ â€” 5 = z, and Va â€”\/ â€” b =â–  v ; also put z -^ v â– =. 1x.
Hence it will be 1st, z* = a -f y^â€” i, and 2nd, \?=.aâ€”i^â€”b; conse-
quently 7? -\- %? â– =. la. But since z + w =: 1x, therefore ^ ^ = - =: zz â€”

zv -f- w. But (z + 'vf' = zz -|- 2zi; + w = Axx; therefore Zzv = Axx â€” -.
Now since zV = wa -}- i ; therefore zv = 1/ aa â– \- b; which being put = m,
it will then be Axx â€” - = 3m, or Ax^ â€” Smar = a.

X '

Hitherto we have had two kinds of equations ; the first, in which m was
put = ^/ aa â€” b ; the latter, in which it was ^ l/aa -{â–  b. The former may
be called hyperbolical, the latter circular.

Prob. 2. â€” To extract the Cubic Root of the Impossible Binomial a -\- s^ â€” b.
Suppose that root to be x + v' â€” y, the cube of which is x^ -j- 3xx\^ â€” y

â€” ^^y â€” yVâ€”y-

Now put a^ â€” 3xy = a, and 3xxÂ»/ â€” y â€” yi/ â€” y â– =â– Â»/â€” b. Then the
squares of these will give two new equations, viz.
x^ â€” 6T*y + ^x^y"^ = aa,

â€” ^x^y + Qx'^y^ ~ xf â– = â€” b.

Then the difference of these squares is .r" + 3,r*y + ^x^y* -\- if â– = aa â€˘\- b\
the cubic root of which \^ xx -\- y =. s/ aa -^ b â€˘=â–  m suppose ; hence y = m
â€” XX ; which value of ^ substituted in the equation o^ â€” 3xy = a, gives x^ â€”
Smx -j- 3x^ = a, or 4x^ â€” 3mx =. a ; which is the very same equation, as
had before been deduced from the equation 2x = 1/ a -\-'/ â€” b -^-V a â€”Â»/ â€”b\
but yet it does not follow, that in the equation Ao^ â€” 3mx = a, the value of ar
can be found by the former equation, since it consists of two parts each in-
cluding the imaginary quantity ^/ â€” b ; but this will be best done by means of
a table of sines.

Therefore let the cube root be extracted of the binomial 81 -f i/â€” 2700.
Put a = Ql, b = 2700: then aa + b = 656i + 2700 = 9261, the cubic

N N 2

276 PHILOSOPHICAL TRANSACTIONS. [aNNO J 738.

root of which is 21, which put = m, which makes 3mx = 63x; therefore the
equation to be resolved will be 4a;' â€” 63x = 81, which being compared with
the equation for the cosines, viz. 4x^ â€” 3rrx = rrc, gives rr = 21, hence

r a 81 27

r = ^1\, therefore c = â€” = â€” = -y.

27
To find then the circular arc to the radius \/1\, and cosme â€” ; put the

whole circumference = c, and take the arcs |, ^^, ^-~^, which will easily
be known by a trigonometrical calculation, especially by using logarithms; then
the cosines of the arcs to the radius \/21, will be three roots of the quantity x;
and since y = m â€” xx, there will therefore be as many values oiy, and thence
a triple value of the cube root of the binomial 81 + â€˘/ -r- 2700 ; which must
now be accommodated to numbers.

Make then as y/1\ : V - so is the tabular radius : to the cosine of an arc a,
which will be nearly 32Â° 42' ; hence the arc c â€” a will be 327Â° 18', and c + a
392Â° 42', of which the 3d parts will be 10Â° 54', and 109Â° 6', and 130Â° 54'.
But now as the first of these is less than a quadrant, its cosine, that is, the sine
of 79Â° 6', ought to be considered as positive ; and both the other two being
greater than a quadrant, their cosines, that is, the sines of the arcs 19Â° 6' and
40Â° 54', must be considered as negative. Now by trigonometrical calculation
it appears, that these sines, to radius \/21, will be 4.O4999, and â€” I.4999,
and 3.0000, or f, and â€”^, and â€”3. Hence there will be as many values of
the quantity y, viz. all those represented by ot â€” xx, viz. 21 â€” V> and 21 â€” -Â«-,
and 21â€”9, that is, 4-, VS '2, the square roots of which are i\^3, 4-/ 3,
2^3 ; therefore the three val ues of .y/ â€” 3/, wi ll be -l-/ â€” 3, â– |-v' â€” 3, 2./ â€” 3 ;
hence the three values of ^81 -\- V â€” 2700 are 4 + 4.^â€”3, and â€”4+4

^ 3j and â€” 3 + iy/ â€” 3. And by proceeding in the same manner, there

will be found the three values of v^81 â€”Vâ€” 270O, which are 4.-4.^â€”3,
and f â€” 4/â€” 3, and â€” 3 â€” -i-v^ â€” 3.

There have been several authors, and among them Dr. Wallis, who have
thought that those cubic equations, which are referred to the circle, may be
solved by the extraction of the cube root of an imaginary quantity, as of 81 +
V^â€” 2700, without any regard to the table of sines : but that is a mere fiction;
and a begging of the question ; for on attempting it, the result always recurs
back again to the same equation as that first proposed. And the thing cannot
be done directly, without the help of the table of sines, especially when the
roots are irrational ; as has been observed by many others.

Prob. 3. â€” To extract the nth Root of the Impossible Binomial a + \/ â€” b.

Let that root he x -^ V â€” y; then making V aa -{â–  b = m, and â€” â€” =/>,

VOL. XL.3 PHILOSOPHICAL TRANSACTIONS. 1'J'J

describe, or conceive to be described a circle, tiie radius of which is v'/ra, in
which take any arc a, the cosine of which is â€”r\ and let c be the whole cir-
cumference. To the same radius take the cosines of the arcs

AC â€” AC+a2C â€” a2C+a3C â€” ASC+A. -hi 1 r ^

-, , , , , , , &c. till the number or them

be equal to n. Then all these cosines will be so many values of x ; and the
quantity y will always be m â€” xx.

Prob. 4. â€” Having given any Equation, of the Kind of those above described;
to know whether its Solution is to be referred to the Hyperbola or to the Circle.

Let n denote the highest dimension of the equation : divide the coefficient of
the second term by 2""' X n, calling the quotient m : then see whether the
square aa be greater or less than m"; if it be greater, the equation is to be re-
ferred to the hyperbola ; but if less, to the circle.

Let there be given the equation l6,r* â€” 40:1^ + 20x = 7, where n := 5 ;
therefore 2""' X Â« = 20: divide 40 by 20, the quotient is 2 = ?Â«; hence m" =.
32, and aa := 40; and as this is greater than the power 32, the equation is to
be referred to the hyperbola. But since in the hyperbolical case there was put
i/ aa â€” b = m, it follows that aa â€” b = m^ = 32, and therefore b = aa â€” 32
= 49 â€” 32 = 17. Now the root of the equation in this case is ^v'7 -)_^ jy
^7 _^ 17 : but v^l? = 4.123105 nearly; therefore 7 +/17 = 11.123105,
and 7â€”^17 = 2.876895 ; also the 5th root of the former number is J.6221,
and the 5th root of the latter 1.2353, the sum of which roots is 2.8574, and
the half sum 1.4287 is the value of x in the given equation.

Again, let the equation 16a;* â€” 40a^ + 20^ = 5 be given ; in which m is
\ still = 2, but a = 5, and the square aa is less than 2^ or 32 ; therefore the
value of X cannot be obtained without the quinquisection of an angle ; and that
is performed by our general theorem, by taking, to the radius i/2, the arc
whose cosine is ^ = ^ = -, which is the arc of 27Â° 55' nearly, the 5th part
of which is 5Â° 35'. Now the log. cosine of that arc, to the radius l, is
9.9979347 ; but since our radius is V2, to that log. add the log. of ^2, that
is 0.1515150, the sum will be 10.1484497; from which taking away the 10,
the remainder 0.1484497 will be the log. of 1.4075 nearly, the number sought.
And in like manner the other four roots may be found.

It may be further remarked, that if the equation be of the hyperbolic kind,
and n be an odd number, there will be only one possible root ; but if n be an
even number, there will be only one value of the square xx, the rest being
impossible.

If the equation be of the circular kind, all the roots will be possible.

To know how many of the roots will be affirmative, and how many negative,

378 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739.

in equations to cosines, observe the following rule: if n be an even number,
there will be as many affirmative roots as negative. But if n be an odd num-
ber, but such that â€” ^ be an even number, the number of affirmative roots

will be ^^^^^, and the number of negative â€” ^.

But if ^ be an odd number, the number of affirmative roots will be "\-,

and the number of negative

n â€” 1

2
END OF THE FORTIETH VOLUME OF THE ORIGINAL.

A Catalogue of the Fifty Plants from Chelsea Garden, presented to the Royal
Society by the Company of Apothecaries, for tfw Year 1737, pursuant to the
Direction of Sir Hans Sloane, P. R. S. By Isaac Rand, F. R. S. N" 432,
p. 1. Vol.XLL

This is the l6th annual presentation of this kind, making the number of
800 different plants. ^

Of the Measure and Motion of Effluent Water. By James Jurin, M. D. F. R. S.
^c. NÂ° 452, p. 5. From the Latin.

Essay I. Of Water issuing from a Vessel kept always full, through a
round Hole ; and of its Resistance arising from a Defect of Lubricity. â€” ^The
ancients had no other measure of effluent water, than that uncertain and falla-
cious one, which, having no regard to the velocity, depended wholly on the
perpendicular section of the stream. The first who opened a way to the truth,
was Castelli, an Italian, and the friend of Galileo. He, having discovered that
the quantity of water flowing through a given section of a stream, is not
given, as the ancients thought, but that it is proportional to the celerity with
which the water is carried through it ; by this noble discovery he laid the foun-
dation of a new and most useful hydraulic science. This discovery therefore
engaged the philosophers to study this doctrine so carefully, that after Castelli's
time there was hardly any eminent mathematician, who did not endeavour to
add something to it, either by experiments, or by reasonings and argument*
k priori.

But most of them, notwithstanding their great abilities, had no success in
it, because of the exceeding difficulty of the work. For those who studied
only the theory, laid down such theorems as were found to be false, when

VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 279

brought to the test by experiments; and those who laboured in making experi-
ments, omitting to observe some minute circumstances, the importance of
which they had not yet perceived, differed greatly from one another, and al-
most all of them erred from the real measure.

Of this there cannot be given a better example, than that simple and easy
one, which has generally been a foundation for all the rest, and is what we
have now undertaken to treat of diligently, when water issues through a circu-
lar hole made in the bottom of a vessel constantly full, with a constant velocity,
Poleni alone has given the true measure of the water flowing out, or at least
very near the true one; and Sir I. Newton alone has laid the foundation of
discovering that measure; though most have rejected it, and some, concealing
the author's name, have pretended that it was their own.

We shall therefore make our attempt under the conduct of these two leaders;
and in the first place propose, under the name of phaenomena, such things as
either appear from experiments, or are confirmed by certain reasonings drawn
from them; and in the last place, we shall attempt the solution of those phse-
nomena.

Phenomena of Water Rowing through a Hole in the Bottom of a Vessel
constantly full. â€” 1. The depth of the water, and the time of flowing out being
given, the measure of the effluent water is nearly in proportion to the hole.

1. The depth of the water, and also the hole being given, the measure of
the effluent water is in proportion to the time.

3. The time of flowing out, and the hole being given, the measure of the
effluent water is nearly in a subduplicate proportion to the height of the water.

4. The measure of the effluent water is nearly in a ratio compounded of the
proportion of the hole, the proportion of the time, and a subduplicate propor-
tion of the depth of the water.

5. The measure of water flowing out in a given time, is much less than that
which is commonly assigned by mathematical theorems. For the velocity of
effluent water is commonly supposed to be that which a heavy body would
acquire in vacuo by falling from the whole height of the water above the hole,
and this being supposed, if we call the area of the hole f, the height of the
water above the hole a, the velocity which a heavy body acquires by falling in
vacuo from that height v, and the time of falling t, and if the water flows out
with this constant velocity v, in the time t; then the length of the column of
water, which flows out in that time, will be 2a ; and the measure of it will be
2ap. But if we calculate from the most accurate experiments of Poleni, we
shall find the quantity of water which flows out in that time, to be no more
than about v*oVo of this measure 2af.

280 PHILOSOPHICAL TKANSACTIONS. [anNO 1739-

Poleni's experiments seem to be preferable to all others, not only because of
his extraordinary diligence and accuracy, but on other accounts also. He
found, that the quantity of water flowing out of a vessel through a cylindrical
tube, far exceeded that which flowed through a circular hole made in a thin
plate, the tube and hole being of equal diameter, and the height of the water
over both being also equal. And he found it to be so, when the tube was in-
serted, not only into the bottom, which others had observed before, but also into
the side of the vessel.

Now a hole made in the thinnest plate must be considered as a short cylin-
drical tube. Whence it appears that a greater quantity of water runs through
a hole made in a thin plate, than would have run out, if the thickness of the
plate had been what is called infinitely small. But as such a plate can neither
exist, nor even be conceived by the imagination, it remains that we increase
the diameter of the hole, that the thickness of the plate may bear the least
proportion possible to that diameter.

This Poleni performed with great judgment, when he made use of a diameter
of 26 lines, and not quite a line thick ; whereas before him hardly any one
made use of a diameter of above 6 or 7 lines, or ever attended to the thickness
of the plate or bottom of the vessel, except Sir I. Newton, who mentions his
making use of a very thin plate. But Poleni exceeded all others, in consider-
ing not only the size of the hole, but of the vessel also, that the water might
descend toward the hole with the greatest freedom, and the least impediment ;
so that there can be no doubt but that the measures taken by him, come much
nearer the truth than any other.

6. Since then the measure of the water running out in the abovementioned
time T, is 2af X toVs-* the length of the column of water, which runs out
in that time, is 2a X -nrgV- Therefore if each of the particles of water,
which are in the hole in the same space of time, passes with equal velocity, it
is plain that the common velocity of them all, is that which the space 2a X
would be gone over in the time t, or the velocity v x , Vo'o â€˘ But this

is the velocity with which water could spring or jet in vacuo to near ^ of the
height of the water above the hole.

7. But when the motion of water is turned upwards, as in fountains, these
are seen to rise almost to the whole height of the water in the cistern. There-
fore the water, or at least some portion of it, spouts from the hole with
almost the whole velocity v, and certainly with a much greater velocity than

V y\ 1 Â« â€˘

8. Hence it is evident, that the particles of water which are in the hole at
the same point of time, do not all burst out with the same velocity, or they

vol.. 'XLI.] PHILOSOPHICAL TRANSACTIONS. 28)

have no common velocity. Though mathematicians have hitherto taken the
contrary to be certain.

g. At a small distance from the hole, the diameter of the vein of water is
much less than that in the hole. For instance, if the diameter of the hole be
J , the diameter of the vein of water will be 44 or 0.84, according to Sir I. New-
ton's measure, who first observed this wonderful phenomenon ; but according to
Poleni's measure -|4 or '-/-r* ; '^hat is, if you take the mean diameter, 0.78 nearly.

We should now proceed to the solution of these phaenomena ; but before
doing this, it wiil be convenient to notice the following particulars.

1. We consider water no otherwise than as a fluid and continuous body,
the parts of which yield to the least force, and are thereby moved among
themselves.

2. By efRuent water, is understood that quantity of It, which actually passes
out of the hole; and though it may seem unnecessary, yet it may he proper to
mention, that in my dissertation on the motion of running waters, inserted
about 24 years ago, in the Philos. Trans, by defluent water I understood that
whole quantity of water, which is put in motion within the vessel, and descends
towards the hole. jirr.

3. We consider the amplitude of the vessel as infinite, or at least so great,
that the decrease of the depth of water, during the whole space of time in
which the water flows out of the hole, is imperceptible.

4. We consider water as running out with a constant velocity. At the be-
ginning indeed of the motion it runs out, for a very small space of time, with
a less velocity than afterwards. But we pass over the very beginning of the
motion, and investigate the measure and motion of water, when it has acquired
its utmost velocity. Now this must necessarily be constant, as long as the
height of the superincumbent water remains the same.

5. We conceive the bottom of the vessel no otherwise than as a mathe-
matical plane, or at least as so thin a plate, that its thickness is little or nothing,
with regard to the diameter of the hole.

6. By the measure of effluent water in the following pages, we always under-
stand that quantity of water which flows out of the hole in the same space of
time that a heavy body, falling in vacuo, would take in passing through the
height of the water above the hole. ,

7. By the motion of effluent water, we understand the sum of the motions
of all the particles of water, which run out of the hole in the abovementioned
space of time. But the motion of every particle^ is as the factum of the par-
ticle itself, and of the velocity with which it bursts out of the hole.

8. That what we shall say hereafter may be the more easily ajnceived, we

VOL. VIII. O o

282 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739-

shall first propose the more simple cases, and then proceed to those which are
more compound, but nearer to the true state of things. Thus, in the first
Problem, that the solution may be the more simple, we suppose the water to
run out of the hole into a vacuum, and the particles of water, while they
descend towards the hole, to be without any resistance arising from a defect of
lubricity.

In the 2d and 3d Problems, the efflux of the water is still supposed to be
in vacuo, but we conceive the particles of water, while they descend towards
the hole, to meet with some resistance for want of lubricity, but so small, that
the decrease of the motion of the water running out of the hole, thus occa-
sioned, is to be accounted as nothing.

In the 4th and 5th we still retain the supposition of the vacuum ; but the
decrease of the motion of the effluent water, for want of lubricity, is supposed
to be sensible.

Lastly, in the 6th and following Problems, we consider the thing as it really
is, when it is transacted in the air, so that the particles of water suffer a sensi-
ble resistance, not only from each other for want of lubricity, within the vessel,
but also after their going out of the vessel, from the attrition of the am-
bient air.

Pkob. 1 . To determine the Motion, Measure, and Velocity of IVater running
into a Vacuum, through a Hole in the Bottom of a Vessel, where the Par-'
tides of IVater meet with no Resistance for want of Lubricity. â€” So long as
the hole is stopped, the stopper sustains the weight of a column of water
lying perpendicularly over it. On removing the stopper, the column of water,
which lies perpendicularly over it, being no longer sustained, by its pressure
causes the water to run out through the hole, and after having brought it to
its due velocity, keeps the velocity of the effluent water constant, by its con-
stant pressure.

, It must be conceived indeed, that the motion of the water running out of
the hole is derived, not only from the weight of the perpendicular column, but
partly from the pressure of this column, and partly from the pressure of the
surrounding water. But this makes the motion of the effluent water neither
greater nor less, than if it arose from the pressure only of the perpendicular
column : not less, because the pressure of the perpendicular column, if it is
obstructed, will generate a motion proportionable to itself, and it can only be
hindered so far as the surrounding fluid urges the effluent water : not greater,
because the pressure of the surrounding fluid can add nothing to the motion
of the effluent water, unless it takes away as much from the pressure of the
perpendicular column.

. 1 A ' ',

rOL. XLI.] PHILOSOPHICAL TRAN8A.CTIOK8. 283

Therefore the adequate motion of the water, flowing out of the hole, is tlie
pressure or weight of the column of water over the hole. But a given force,
howsoever applied, generates a given quantity of motion in a given time, to-