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Suppofe x (AE) to be increas'd by the indefinitely - little

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2 x 3 (=AFFE) will become

— 2¢ a3] = 4 r2 x2 - 2 r x 3 — 6 r x2a + 8 r2xa-6rxa2 + 4 r2 a2 (= AN× PN ;) but the former being a Maximum, is therefore greater than the latter; and confequently the Square of the former, viz. 472x2 - 2x is, the Square of the latter, viz. 4rx— 2 rx 3 — 6 rx 2 4 + 8 r2 xa — 6ṛxa2 +4r2a2-2ra3. Wherefore 6rxx+6rxa — 472 a → 58rax.

2

Again, fuppofex (AE) to be diminish'd by the indefinitely-little

Quantity 4, (EP) then 47 x27x3(AF x FE) will become ÷

4r2x2 - 2rx3+6rx2a

8 r3x a — 6 rx a2+ 4r3a2+ 2 x a3 | (AN× PN:) But the former being, by Suppofition, greater than the latter, the Square of the former is therefore greater than the Square of the latter that is, 4r' x2 - 2rx3 C 4x2x2-2rx3+6rx2a-8rxa — 6rxa2+4r1 a2 + 2 ra3; wherefore 8 rx + 6rxa - 4r2 4 6rx3. oF

From what has been faid it is evident, that ift, 6rxa 472 a; 2dly, (by our Lem. 1.) 8rx=6rx*; and 8r=6x

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Note, If this Queftion had been propos'd, viz. x2 — x being == Maximum; what is x = to?

Here 'tis evident, that by how much the greater you fup-pofe the Value of x to be, by fo much the nearer you come to it; therefore it is not to be determin'd, unless you suppose,

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that Infinity is fufficiently great; then x may be But - :

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if any Number can be affign'd that is greater than, as 2, 3

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or &c. (for really I do not know which are they Equal, or retain the fame Proportion as their Numerators) then the Value of x is not to be affign'd: And after all, a Negative feems to go nearer to give the Value of x than the lame Number if Affirmative: And therefore I have all along exempted Negatives as the very Operations do. But if this Queftion had been propos'd, viz. x —— Minimum; then the Va lue of x will be found (by the foregoing Method) <.

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K kk 2

Propo

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Let C F be a Chord, and let one End thereof be made fast at C, and to the other end faften the Pulley F, abour which fufpend the Weight D by the Chord DF B, faftening one end thereof at B, and let the Points B and C be in the fame Horizontal Line CB: And fuppofe both the Chords and the. Pulley to be without Weight, ris requir'd to find the lowest Defcent of the Pulley, and of the Weight.

*

It is evident, that the Weight D will defçend below the Horizontal Line CB, as low as the Chords CF and BFD will permit, and therefore the Line DFE will reprefent the greatest Defcent of the Weight D.

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Suppofe the known Quantities DFBb, CB c, and CF4: And fuppofe the unknown and variable Quantity

CE=x; EFdx = × ; then is E F — √ √ — x2, and FB = dd+cc - 20 x 1 =

and DFF = b − d ́ + c* — 2{cxp3 + d® imum. Quere, x.

-Max

Note, What is meant by the Chords and Pulleys being without Weight, is, that they are of no Weight in respect of the Weight D: For if that be not fuppas'd, the weight of the Said Chords and Pulley. be a means to hinder the Weight D's defcending as low as the the Chords will permit.

be

Solution

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that is (by Evolution) ba+c2cx1 +

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d

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2x4

- AA

2

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. Which laft

-20x

2dly, Suppose to be diminish'd by the indefinitely - little'

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become b

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d. + c 2cx+2cal 2+d2. x2 + 2x a da But the former is a Maximum, and must therefore be greater

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than the latter; whence it will follow, that

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From the latter Parts of the first and second Suppositions

it is manifeft, that the Difference of

c

Vdd + cc - 20%

and

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

Suppofitions (by our Lem. 1.)

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C

Vdd+cc - 20x Vdd

x

- xx

wherefore ccddccxx ddxx + ccxx - 2cx; that is, 2cx3- 202x2 - d2x2 + c2 do. Finally, by dividing each part by xc you'll have 2 cx2-d'x-cdo.

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Iven the Bafe A C, and Perpendicular A B of the Lẻ
A ABC; required to find its Area.

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Supppofe the I. BA to be divided into an indefinite Number of equal Parts BD, DE, EF, &c. and Z A; and fuppofe each of thofe Partsa; then, through the Points B, D, E, F, &c. and Z, draw the Lines Bb, D'Dd,

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AC, and

Eere, Fff, &c. and Z
thro' the Points D, E, F, &c. and 3. Draw

the Lines be, def, ef, &c.

a and C

BA; then the Sum of the Os Bb DD,

Note, I fuppofe it known, that the Area of any Dis had by length by its multiplying the Perpendicular Height.

DITE, EeFF, &c. and Z CA is called the Sum

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of the Circumfcribing ; and the Sum
t
of the SD De E, ETF, &c. and Z
a A is call'd the Sum of the infcrib'd
Now the first, (or leaft) fecond, third, &c.
fcribing being firft, fecond, third,

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