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That is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.

SCHOLIUM

176. On the foregoing principles depends the nature of scales and beams, for weighing all sorts of goods. For, if the weights be equal, then will the distances be equal als, which gives the construcsion of the common scales, which ought to have these properties :

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proportions, which is the true weight of thiet
oman Statera, or Steepard, is also a

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ward and forward, to different distances, an enger arm of the lever; and it is thus constructed:

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Let Ae be the steelyard, and cis centre of action, r,202 the divisions must commence I the wo each other: if not, slide the constant ovente i along from B towards C, till it ust balance she herri without a weight, and there make a jotch 'e eam. marking it with a cipher . Then hang 2 at 4 : TO equal to 1, and slide I back towards 3 til 2 diance such other; there notch the beam, and mark i Toma make the weight w double of 1, and stiding : 205 calance it, there mark it with 2. Do the same **!, b, si, si making w equal to 3, 4, 5, ke, times 1; rc he near finished. Then, to find the weight of V NY steelyard; take off the weight w, ad an. Tom at A; then slide the weight I backwart 01 just balance the body b, which suppose que 5; then is bequal to 5 times the weight ti. pound, then b is á pounds; jirt Fjelman en 10 pounds; and so on.

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OF THE WHEEL

PROPOSITION 179. In the best and state le

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That is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other.

SCHOLIUM.

C

D

centre B.

176. On the foregoing principles depends the nature of scales and beams, for weighing all sorts of goods. For, if the weights be equal, then will the distances be equal

B also, which gives the construction of the common scales, which ought to have these

E properties :

1st, That the points of suspension of the scales and the centre of motion of the beam, A, B, C, should be in a straight line: 2d, That the arms AB, BC, be of an equal length: 3d, That the centre of gravity be in the centre of motion By or a little below it : 4th, That they be in equilibrio when empty: 5th, That there be as little friction as possible at the

A defect in any of these properties, makes the scales either imperfect or false. But it often happens that the one side of the beam is made shorter than the other, and the defect covered by making that scale the heavier, by which means the scales hang in equilibrio when empty; but when they are charged with any weights, so as to be still in equilibrio, those weights are not cqual; but the deceit will be detected by changing the weights to the contrary sides, for then the equilibrium will be immediately destroyed.

177. To find the true weight of any body by such a false balance:--First weigh the body in one scale, and afterwards weigh it in the other; then the mean proportional between these two weights, will be the true weight required. For, if any body b weighi w pounds or ounces in the scale D, and only w pounds or ounces in the scale E: then we have these two equations, namely, AB b

and b = AB. the product of the two is AB • IC . b = AB • BC . Ww; hence then

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= WW, and

b = vww, the mean proportional, which is the true weight of the body b.

178. The Roman Statera, or Steelyard, is also a lever, but of unequal brachia or arms, so contrived, that one weight only may serve to weigh a great maný, by sliding it back

ward

- BC

.

W.

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W;

ward and forward, to different distances, on the longer arm of the lever; and it is thus constructed :

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Let AB be the steelyard, and cits centre of motion, whence the divisions must commence if the two arms just balance each other: if not, slide the constant moveable weight i along from B towards c, till it just balance the other end without a weight, and there make a notch in the beam, marking it with a cipher 0. Then hang on

Then hang on at A a weight w equal to 1, and slide 1 back towards B till they balance each other; there notch the beam, and mark it with 1. Then make the weight w double of 1, and sliding 1 back to balance it, there mark it with 2. Do the same at 3, 4, 5, &c, by making w equal to 3, 4, 5, &c, times I; and the beam is finished. Then, to find the weight of any body b by the steelyard ; take off the weight w, and hang on the body b at A; then slide the weight I backward and forward till it just balance the body b, which suppose to be at the number 5; then is b equal to 5 times the weight of 1. So, if I be one pound, then b is 5 pounds; but if i be 2 pounds, then b is 10 pounds; and so on.

Of The WHEEL AND AXLE.

PROPOSITION XXXII. 179. In the Wheel-and-Axle; the Weight and Power will be . in Equilibrio, when the Power p is to the Weight w, Reciprocally as the Radii of the Circles where they act; that is, as the Radius of the Axle ca, where the_Weight hangs, to the Radiús of the Wheel CB, where the Power acts. That is, P:W:: CA: CB, HERE' the cord, by which the power p acts, goes about

a

the circumference of the wheel, while that of the weight w goes round its

axle, or another smaller wheel, attach-ed to the larger, and having the same

BAD axis or centre c. So that BÀ is a lever moveable about the point c, the power P acting always at the distance BC, and the weight w at the distance CA; therefore P:W:: CA : CB.

180. Cordl. 1. If the wheel be put in motion; then, the spaces moved being as the circumferences, or as the radii, the velocity of w will be to the velocity of P, as ca to CB; that is, the weight is moved as much slower, as it is heavier than the power; so that what is gained in power, is lost in time, And this is the universal property of all machines and engines.

181. Corol. 2. If the power do not act at right angles to the radius cb, but obliquely; draw cd perpendicular to the direction of the power, then, by the nature of the lever, Piw::CA : CD.

SCHOLIUM. 182. To this power be

B

D long all turning or wheel machines of different radii.

ES Thus, in the roller turning

с on the axis or spindle CE, by the handle CBD;

the power applied at b is to the weight won the roller,

the radius of the roller is to the radius CB of the handle.

183. And the same for all cranes, capstans, windlasses, and such like; the power being to the weight, always as the radius or lever at which the weight acts, to that at which the power acts; so that they are always in the reciprocal ratio of their velocities. And to the same principle may be referred the gimblet and augur for boring holes.

184. But all this, however, is on supposition that the ropes or cords, sustaining the weights, are of no sensible thickness. For, if the thickness be considerable, or if there be several folds of them, over one another, on the roller or barrel; then we must measure to the middle of the outermost rope, for

the

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