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Notes on Book III. to A A. The Square of 6 \ the Half of the Root is 42 4 ; from which fubftradting 32, there remains 10 %, the Root of which is 3^ a little left : Take that from 6 i, and there remains 3 and ; add that to 6 f, and it will make p if; and thefe 3 ^ and if will be the two Roots; which thews that the Water can never rife when the Barrel is empty, above 3 foot and a little more, tho’ you play the Pifton as long as you pleafe ; but if you had fill’d the Barrel 9 Foot i f, you might make the Water rife ia Foot compleat by feveral Strokes of the Pifton. Let us fuppofe now that the Barrel is 14 Foot up to the Pifton, arid that the Stroke of the Pifton is 2 Foot; 32 —’A will be to 32, as 14 — A to 16 A. To find the Equation ealily, you muft multiply 32 by 2, the Difference of 14 and 16 : The Produa is <*4 for the abfo- lute Number, and that of 16 A, will be the Number of the Roots, and A A will be equal to 16 A — <?4; the Square of half the Root is 64; from whence fub- traaing 64, there remains o, whofe Root is o, which being taken from and added to 8, ftill makes 8; which ftiews that there is’but one Root, and that the Water can’t rife above 8 Foot; but if you make the Pillon play ever fo little higher than 2 Foot, the Water will rife 14 Foot. The Analogy is eafy ; for the Pifton being rai- fed 2 Foot, the Barrel will be id Foot, and that Water being at 8 Foot, there will remain 5 boot of Air ; but 32 is to 24 the Complement of 8 Foot to 32, as 8 Foot of rarified Air to 5 Foot of common Air; then the Water will raife no higher than 8 Foot, if the Pifton plays but 2 Foot. Thence you fee, that to draw up Wa ter to a conliderable Height, as 20 Foot, the Breadth of the Pump-Barrel muft be dimi- iiifti’d, and a fufficient Space muft be al low’d for the Stroke of the Pifton; for, luppofing that the Surface of the Pifton be 4 times broader than the Bafe of the Bar rel the riling of the Pifton 1 Foot, will have the fame Eftedt as if it role 4, if the Diameter of the Pifton were only equal to that of the Barrel ; if then the Srroke 1oe a Foot and a half, it will be the lame as if it role 6 Foot, and were of the fame B r 2adth : Now the 4 Terms of Equation being 32 — A; 32, 20 —A, 25^ A, there will be 6 times 32, viz. 192 for one Term of the Equation, and 16 A for the other, according to what has been laid ; there will be then A A equal to 25 A 192 ; the Square of half the Roots is 169 left than 192 ; and confequently if you pump a long time, you may raife the Water 20 Foot. If in the Example above-mention’d, you take 8 Foot for the higheft Term of the Water, when the Barrel is 14 Foot, and the Stroke of the Pifton 2 Foot, ’tis eafy to prove, that if you fuppofe 9 Foot of Water upon the Clack, it will continue to rife by the playing of the Pifton 2 Foot; for there will remain 5 Foot of Air. Now there is a left Proportion betwixt 5 and 7, than there is betwixt 27, the Complement of 5 to 32, and 32, and confequently the Water will rife higher than 9 Foot The Proportion will ftill be more unequal, if you take 10 or 11 Foot; and if you take 7 inftead of 8 Foot, the Water will ftill rile, for there will remain 7 Foot of Air; now 25, the Complement of 7 to 32, is to 32 as 7 to 8 if; then if the Pifton goes 2 Foot, it will raife the Water higher than 7 Foot; it will rife ftill more ealily, if you pour in only 5 Foot of Water; for there will be 8 Foot of Air. Now the Complement 25 is to 32 as 8 to 9 ; then if inftead of 9 which makes the Equi librium, the Pifton goes to Foot, it will make the Water rife Hill better than when it was at 7 Foot; and better ftill than when it is at 5 Foot, £sV. If you would know what Play the Pifton muft have to raife the Water 30 Foot, you muft take a Number a little greater than the half of 50, as id, at which Point pretty near the YVatcr, will rile with the greateft Difficulty ; the Comple ment is 15, the Remainder of Air is 14 ; as 15 is to 32, lb is 14 to 28. The Pifton then muft tile 14 Foot, or if the Barrel be 2 Inches Diameter, the pifton mull be 7 Inches ’ ; for the Square of 7 ; j s ^ which is a little more than 14 times 4 the Square of ailnches; and then it will be fulfi- cient that the Stroke of the Pifton be one Foot; but as it is ftill more difficult at an Elevation of 18 Foot, the Pifton muft be 8 Inches Diameter, to raife the Water above 18 Foot, when its Stroke is but one P'oot. Book IV.