QUEST. 59. With what velocity would each of those three fluids, viz. quicksilver, water, and air, issue through a small orifice in the bottom of vessels, of the respective heights of 30 inches, 35 feet, and 5·5240 miles, estimating the pressure by the whole altitudes, and the air rushing into a vacuum?

Ans. the veloc. of quicksilver 12.681 feet.

QUEST. 60. A very large vessel of 10 feet high (no matter what shape) being kept constantly full of water, by a large supplying cock at the top; if 9 small circular holes, each of an inch diameter, be opened in its perpendicular side at every foot of the depth: it is required to determine the several distances to which they will spout on the horizontal plane of the base, and the quantity of water discharged by all of them in 10 minutes?

√64 8.00000
6.00000

√36

and the quantity discharged in 10 min. 123-8849 gallons.

Note. In this solution, the velocity of the water is supposed to be equal to that which is acquired by a heavy body in falling through the whole height of the water above the orifice, and that it is the same in every part of the holes.

QUEST. 61. If the inner axis of a hollow globe of copper, exhausted of air, be 100 feet; what thickness must it be of, that it may just float in the air?

Ans. 02688 of an inch thick.

QUEST. 62. If a spherical balloon of copper, of of an inch thick, have its cavity of 100 feet diameter, and be filled with inflammable air, of of the gravity of common air, what weight will just balance it, and prevent it from rising up into the atmosphere? Ans. 21273lb.

QUEST. 63. If a glass tube, 36 inches long, close at top, be sunk perpendicularly into water, till its lower or open

end be 30 inches below the surface of the water; how high will the water rise within the tube, the quicksilver in the common barometer at the same time standing at 29 inches? · Ans. 2.26545 inches.

QUEST. 64. If a diving bell, of the form of a parabolic conoid, be let down into the sea to the several depths of 5, 10, 15, and 20 fathoms; it is required to assign the respective heights to which the water will rise within it its axis and the diameter of its base being each 8 feet, and the quicksilver in the barometer standing at 30.9 inches?

Ans. at 5 fathoms deep the water rises 2-03546 feet.

Art. 1. In the Doctrine of Fluxions, magnitudes or quantities of all kinds are considered, not as made up of a number of small parts, but as generated by continued motion, by means of which they increase or decrease. As, a line by the motion of a point; a surface by the motion of a line; and a solid by the motion of a surface. So likewise, time may be considered as represented by a line, increasing uniformly by the motion of a point. And quantities of all kinds whatever, which are capable of increase and decrease, may in like manner be represented by geometrical magnitudes, conceived to be generated by motion. Indeed, notwithstanding all that has been advanced to the contrary, this seems the most natural, as well as the simplest, way of conducting the higher investigations; since it is impossible to conceive a geometrical magnitude to be brought into existence, or to change its magnitude, figure, or place, without motion.

2. Any quantity thus generated, and variable, is called a Fluent, or a Flowing Quantity. And the rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity, at that position or instant: and it is proportional to the magnitude by which the flowing quantity would be uniformly increased

in a given time, with the generating celerity uniformly continued during that time.

3. The small quantities that are actually generated, produced, or described, in any small given time, and by any continued motion, either uniform or variable, are called Increments.

4. Hence, if the motion of increase be uniform, by which increments are generated, the increments will in that case be proportional, or equal, to the measures of the fluxions: but if the motion of increase be accelerated, the increment so generated, in a given finite time, will exceed the fluxion: and if it be a decreasing motion, the increment, so generated, will be less than the fluxion. But if the time be indefinitely small, so that the motion be considered as uniform for that instant; then these nascent increments will always be proportional, or equal, to the fluxions, and may be substituted instead of them, in any calculation.

m

P P

5. To illustrate these definitions: Suppose a point m be conceived to move from the position A, and to generate a line AP, A by a motion any how regulated; and suppose the celerity of the point m, at any position P, to be such as would, if from thence it should become or continue uniform, be sufficient to cause the point to describe, or pass uniformly over, the distance rp, in the given time allowed for the fluxion: then will the said line p represent the fluxion of the fluent, or flowing line, AP, at that position.

B

C

n

m

P P

D

6. Again, suppose the right line mn to move from the position AB, continually parallel to itself, with any continued motion, so as to generate the fluent or flowing rectangle ABQP, while the point m describes the line AP: also, let the distance rp taken, as before, to express the fluxion of the line or base AP; and complete the rectangle raqp. Then, like as rp is the fluxion of the line AP, so is pq the fluxion of the flowing parallelogram AQ; both these fluxions, or increments, being uniformly described in the same time.

7. In like manner, if the solid AERP be conceived to be generated by the plane PQR, moving from the position ABE, always parallel to itself, along the line AD; and if Pp denote the fluxion of the line AP: Then, like as the

E

B

R

be

Σ 19

Α

P P

rectangle Paqp, or PQ XPp, denotes the fluxion of the flowing rectangle ABQP, so also shall the fluxion of the variable solid, or prism ABERQP, be denoted by the prism PQRrqp, or the plane PR X PP. And, in both these last two cases, it appears that the fluxion of the generated rectangle, or prism, is equal to the product of the generating line, or plane, drawn into the fluxion of the line along which it moves.

8. Hitherto the generating line, or plane, has been considered as of a constant and invariable magnitude; in which case the fluent, or quantity generated, is a rectangle, or a prism, the former being described by the motion of a line, and the latter by the motion of a plane. So, in like manner, are other figures, whether plane or solid, conceived to be described by the motion of a Variable Magnitude, whether it be a line or a plane. Thus, let a variable line pq be carried by a parallel motion along AP; or while a point p is carried along, and describes the line AP, suppose another point

AP.

A

to be carried by a motion perpendicular to the former, and to describe the line PQ: let pq be another position of PQ, indefinitely near to the former; and draw or parallel to Now in this case there are several fluents, or flowing quantities, with their respective fluxions; namely, the line or fluent AP, the fluxion of which is pp or ar; the line or fluent rq, the fluxion of which is rq; the curve or oblique line AQ, described by the oblique motion of the point a, the fluxion of which is aq; and lastly, the surface APQ, described by the variable line PQ, the fluxion of which is the rectangle Parp, or PQ X Pp. In the same manner may any solid be conceived to be described, by the motion of a variable plane parallel to itself, substituting the variable plane for the variable line; in which case the fluxion of the solid, at any position, is represented by the variable plane, at that position, drawn into the fluxion of the line along which it is carried.

9. Hence then it follows in general, that the fluxion of any figure, whether plane or solid, at any position, is equal to the section of it, at that position, drawn into the fluxion of the axis, or line along which the variable section is sup.

posed to be perpendicularly carried; that is, the fluxion of the figure AQP, is equal to the plane rQ X rp, when that figure is a solid, or to the ordinate PQ X rp, when the figure is a surface.

10. It also follows from the same premises, that in any curve, or oblique line AQ, whose absciss is AP, and ordinate is PQ, the fluxions of these three form a small right-angled plane triangle aqr; for or = Pp is the fluxion of the absciss AP, qr the fluxion of the ordinate ro, and aq the fluxion of the curve or right line AQ. And consequently that, in any curve, the square of the fluxion of the curve, is equal to the sum of the squares of the fluxions of the absciss and ordinate, when these two are at right angles to each other.

11. From the premises it also appears, that contemporaneous fluents, or quantities that flow or increase together, which are always in a constant ratio to each other, have their fluxions also in the same constant ratio, at every position. For, let AP and BQ be two contemporaneous fluents, described in the same time by the motion of the points P and a, the contemporaneous positions being P, Q, and p, q; and let AP be to BQ, or Ap to вq, constantly in the ratio of 1 to n.

therefore, by subtraction, n X pp = Qq; that is, the fluxion

A

P

B

Q 9

Pp: fluxion oq: : 1 : n,

the same as the fluent AP: fluent BQ:: 1:n,

or, the fluxions and fluents are in the same constant ratio. But if the ratio of the fluents be variable, so will that of the fluxions be also, though not in the same variable ratio with the former, at every position.

NOTATION, &c.

12. To apply the foregoing principles to the determination of the fluxions of algebraic quantities, by means of which those of all other kinds are assigned, it will be necessary first to premise the notation commonly used in this science, with some observations. As, first, that the final letters of the alphabet z, y, x, u, &c. are used to denote variable or flow. ing quantities; and the initial letters, a, b, c, d, &c. to denote constant or invariable ones: Thus, the variable base AP of the flowing rectangular figure ABQP, in art. 6, may be repre