The science of levers: potential and real arms

Figure with letters.

To the left of the figure:

First.

The junction of the lever with its counter-lever will always be right-angled.

In the calculations of mechanical powers no account is taken of the weight or of the position of their material members, but only of the potential lines, that is, the mathematical ones.

I wish to know what weight the heavy body gives of itself to its two supports, namely m q and n b. I shall draw the two potential levers b c, q r, which will be in right-angled junction with b n, m q at the points b, q. Then I shall draw the straight line b q from b to q; and at the middle of these the perpendicular f n will descend below the centre of the heavy body a, and thus I shall have made the right angle b f n at the point f. Besides this, I shall draw the pendant c d in right-angled junction to the lever b c, and so I shall have made the lever b c with its pendant c d, against the potential counter-lever b f with its pendant f n. And if the adversary should say that this counter-lever is not potential but real, that is the counter-lever b n, one answers here with the tenth [proposition] of this, where it says that all the pendants directed toward the centre of the world are in right-angled junction with the arms of the balances placed in the site of equality; and by the first above, every junction of a pendant to the arm of the lever is right-angled, which cannot be, for the adversary, in the junction of b n with the pendant f n. Therefore the right angle b f n serves our purpose, etc.

In the right margin of the sheet, to the right of the figure:

Here one calculates the mathematical powers, and not the real weights, that is, the weights of the members of the instruments.

Calculation.

The proportion that the potential lever b c has to the real counter-lever b f, such will the power of the angle c have to the resistance of the angle f. Therefore, the powers being converse, b c, which is sub-double (half) of b n, will require double power to sustain b n.

In the lower half of the sheet, at the centre, figure with letters.

To the left of the figure:

The power figured here below has similarity with the one above. But this one accounts for the weight that the cord a b c of the heavy body s sustains, that is, the natural weight, called gravity, and the accidental weight, called force; and it teaches how to divide gravity from force. And this operation is done by those cords a b c below the line of equality a d c. But above, one accounts for the weight and force that the heavy body a, or say n, gives of itself, above the line of equality b f q, by the two beams b n, m q, supporters of the said heavy bodies a, n.

Below the figure:

Always the lever and the potential pendant, a e and e b, will be found in right-angled junction at the point e, in the same line as the real pendant b c; that is, continuing the real cord c b in a continuous straight line, it will meet the potential lever a e at the point e in right-angled junction, and the potential counter-lever of the lever a e will be found on the line a d, terminated by the potential pendant d b upon the real pendant b s.

In the right margin of the sheet, to the right of the figure:

Calculation.

The proportion that the lever a e has to the counter-lever a d is here the same as force to weight; but it is converse, because the counter-lever is double the lever, etc...

Therefore the calculation of half the weight s will be found in the potential lever and counter-lever a e and a d, of which the pendants e c and d s are semi-real, as is seen here.

Figure with letters and numbers.

Figure with letters and numbers:

Here 4 of counterweight on the pulley n will move 4 of weight on the pulley m, the axles of such pulleys being of equal thickness and the wheels of various diameters and of equal thickness.

Figure with letters:

Here the real cord m e ends in the potential cord b c, which terminates in the potential arm a b, and the real arm a n is transformed into the potential arm a d.

Figure with letters:

Let there be given the centre of gravity that the weight c has, which is 1 upon the rod a b and is suspended by the cords a d on one side, and f e, c b on the other.

0116 v

Figura con lettere.

A sinistra della figura:

Nel margine destro del foglio, a destra della figura:

Nella metà inferiore del foglio, al centro, figura con lettere.

A sinistra della figura:

Sotto la figura:

Nel margine destro del foglio, a destra della figura:

0119 r

Figura con lettere e numeri.

Figura con lettere e numeri:

Figura con lettere:

Figura con lettere:

Figure with letters.

To the left of the figure:

In the right margin of the sheet, to the right of the figure:

In the lower half of the sheet, at the centre, figure with letters.

To the left of the figure:

Below the figure:

In the right margin of the sheet, to the right of the figure:

Figure with letters and numbers.

Figure with letters and numbers:

Figure with letters:

Figure with letters:

The science of levers: potential and real arms

On this page

Lever and counter-lever meet at a right angle

Mathematical lines, not real weights

Dividing gravity from accidental force

Equal axles, equal loads on pulleys