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Let AB be the Excess of the Diagonal of a Square above its Side, whose Length is to be found.
PROPOSITION X. .
Let AB be the Line to be so divided, that the Rectangle composed of the whole Line and of one of its Parts, shall be equal to the Square formed upon the other Part.
Pon the other pats Parts, T
AB Will be divided at According to the Proposition; for if you make the Rectangle AH, composed of the Line AB, and of the Part BE, it will be equal to the Square AF, formed upon the other Part AE,
PROPOSITION XI. To divide a Right Line of a determined Length, according to given Proportions.
Let AB be a Line proposed to be divided according to the Proporțions C, D, E, F,
P, O, N.
PROPOSITION XII. Upon a Right Line given, to form two Rectangles according to a given Proportion.
AB is the Line upon which two Rectangles are to be formed, which shall in themselves be according to the Proportion of C and D.
ABHF B.2 P. 3.
For the Rectangle
THE following Example of the Nature and Uses of the
1 Mechanic Powers, will not, perhaps, be thought unnecessary, or at least, not improper in this place. .
Mechanics is a mix'd mathematical Science, which confiders. Motion and moving Powers, their Nature and Laws, with the Effects thereof, in Machines, &c.
That Part of Mechanics which confiders the Motion of Bodies arising from Gravity, is by fome called Statics.
Mechanical Powers, denote the fix fimple Machines ; to which ali others, how complex foever, are reducible, and of the Assemblage whereof they are all compounded.
The Mechanical Powers, are the Balance, Lever, Wheel, Pully, Wedge, and Skrew.
They may, however, be all reduced to one, viz. the Lever.
The Principle whcrcon they depend, is the fame in all, and may be conceived from what follows.
The Momentum, Impetus, or Quantity of Motion of any Body, is the Factum of its Velocity, (or: the Space it moves in a giverr Time,) multiplied into its Mals. Hence it follows, that two unequal Bodies will have cqual Moments, is the Lincs they describe be in a reciprocal Ratio of their Maffes.—Thus, if two Bodies, fastened to the Extremities of a Balance or Lever, be in a reciprocal Ratio of their Distances from the Point; when they move, the Lines they defcribe will be in a reciprocal Ratio of their Mafies. For Example.
E, If the Body A be triple the Body B; no and each of them be fixed to the Ex- ,
tremities of a Lever A B, whose Ful
crum, or fixed Point is C, as that the W B Distance of B C be triple the Distance
· CA; the Lever cannot be inclined on either fide, but the Space B E, pailed over by the less Body, will be triple the Space A D, passed over by the great one.
So that their Motions or Moments will be equal, and the two Bodies in æquilibrio.
Hence that noble Challenge of Archimedes, datis viribus, datum pondus movere ; for as the Distance CB may be increased infinitely, the Power or Moment of A may be increased infinitely.--So that the Whole of Mechanics is reduced to the following Problem.
Any Body, as A, with its Velocity C, and also any other Bady, as B, being given ; to find the Velocity necessary to make the Moment, or Quantity of Motion in B, equal to the Moment of A, the given Body.-Here, since the Moment of any Body is equal to the Rectangle under the Velocity, and the Quantity of Matter; as BAC are proportional to a fourth Term, which will be c, the Celerity proper to B, to make its Moment equal to that of A. Wherefore in any Machine or Engine, if the Velocity of the Power be made to the Velocity of the Weight, reciprocally as the Weight is to the Power ; such Power will always sustain, or if the Power be a little increased, move the Weight.
Let, for Instance, A B be a Lever, whose Fulcrum is at C, and let it be moved into the Position a C 6.-Here, the Velocity of any point in the Lever, is as the Distance from the Center. For let the Point A describe the Arch A a, and the Point B the Arch B.b; then these Arches will be the Spaces described by the two Motions : but since the Motions are both made in the same Time, the Spaces will be as the Velocities. But it is plain, the Arches A a and B b will be to one another, as their Radii AC and A B, because the Sectors A Ca, and B Cb, are similar : wherefore the Velocities of the Points A and B, are as their Distances from the Center C.
Now, if any Powers be applied to the Ends of the Lever A and B, in order to raise its Arms up and down; their Force will be expounded by the Perpendiculars S a, and b N; which being as the right Sines of the former Arches, b B and a A, will be to one another also as the Radii A C and C B; wherefore the Velocities of the Powers, are also as their Distances from the Center. And since the Moment of any Body is as its Weight, or gravitating Force, and its Velocity conjunctly; if different Powers or Weights be applied to the Lever, their Moments will always be as the Weights and the Diftances from the Center conjunctly.-Wherefore, if to the Lever, there be two Powers or Weights applied reciprocally proportional to their Distances from the Center, their Moments will be equal; and if they act contrarily, as in the Case of a Stilliard, the Lever will remain in an horizontal
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