I

722

2

a

X • And

Secondly, That is (by our Lemma 1.) =

232

; con

Note, In thefe Operations be fure to take care that your Multiplicator be alwayso, otherwise the Majority will not bold, as in the foregoing Step: Thus, If ab, and you multiply cach by c; if co, then cacb; but if co,then cacb.

D

CHA P. II.

De Maximis & Minimis.

Propofition I.

Ivide into two fuch Parts, that their Product will be a Maximum.

Suppofex

and then b

ft, Suppose the Part x to be increafed by the indefinitely little, or lefs than any affignable Quantity a, then the other

-2xa+ab

a, and xa x

But because bxxx, is (by Suppofition) a Maximum, xxbx-xx − 2 x a + ab

therefore bx

aa:

2dly, Suppose the part x to be diminish'd by the indefinitely fmall Quantity a, then the other part of b will be bx + α ż

-ba

But fince bxxx is (by Suppofition) a Maximum ; there fore bx xxxb x x + 2 x n 02 x A ba- aa, and be 52x.

Now it appears, by the latter parts of the firft and fecond Suppofitions, that, let a be ever fo fmall a Quantity (provided) it be more than nothing) if it be added to 2 x, the Sum will be greater than b; but if it be added to b, then the Sum will be greater than 2 x; confequently (by our Lemma 1.)

Propofition II.

Divide b into three fuch Parts, as being multiply'd together fhall produce a Maximum.

Suppose bx to be one of the Parts requir'd, then x will be the Sum of the other two Parts; but the greatest Product that can be made by any two parts of x, is (by the fore

going Proposition) = — × —; the Question propos'd is there

fore reduc'd to this, viz.

2

being an affirmative Quantity.

ift,Supposer to be encreas'd by the indefinitely-littleQuantity,

the foriner being a Maximum, is therefore greater than the latter;

bx2-x3+2 bx a+ba*—3x2a — 3 × a2 — q3

4

2 b x a + b a 2 — 3 x 2 d — 3 xa-3: Ánd by

4

multiplying each part by, and tranfpofing) 3x2 + 3x4

baib *.

a

2dly, Suppofe x to be diminish'd by the indefinitely fmall

a: But the former being a Maximum, is therefore

4

2

fequently 2 bx +z ± a − b a 3x'; whence, and from bae3x'; the latter Part of the firft Suppofition, 3 xa ba; and 2 bx is (by our Lemma 1) = 32; wherefore 2b = 3x;

Propofition III.

If 1⁄2 be — m, and x" - x" = Maximum; It is requir'd to find the Value of x, it being affirmative.

Suppose x to be augmented by the indefinitely - little Quan

m

tity a: Then x" -x" will become x + al

m

the former being a Maximum, is therefore greater than the latter; that is (by Sir Ifaac Newton's Theorem) x + 5

n

-x

n-i

nx

Wherefore om x

- I

And (by dividing each part by 4, and tranfpofing) nx

Suppofe x to be decreas'd by the indefinitely little Quar

tity 4; then x"

n

x" will become x - 4

4: But

the former being a Maximum, is greater than the latter, that

From the latter parts of the firft and fecond Suppofitions it

Hence, and from the faid latter Parts of the first and se

I

Of all the Cones that can be infcrib'd in a Given Sphere; tis requir'd to find that which has the greateft Convex Surface.

For, if we imagine the Semicircle A FB to revolve about s-Axis AB, it is evident that the Semicirole defcribes a Sphere, and the Lds AFE and ANP defcribe Cones infcrib'd in the fame Sphere, whofe Surfaces are proportional to the respective Rectangles A F× FE, and AN × NP.

14

Suppofe the unknown Quantity AEx, AB. (the Dianerer of the Sphere) =2 And the indefinitely - fhortLine PEEP 4: Then (by the Property of a Circle) = EF, and 24 x FA; and 2 rx] x 2 x 27x3 Maximum, Quere x, it befag an affirmative Quantity.

21x

x x

= 4

Suppofe