Balances analysed by right-angled pendants and arms

Figure with letters.

Beside the figure:

Where the real pendant a b cuts, with its real or potential line, the beam of the balance at right angles, there is generated a space interposed between that cut and the centre of the balance, which is the true length of that arm of the balance. Therefore the semi-real pendant c b, cutting the arm of the balance d f at the point c, makes that arm f d remain in the length c d; and thus I intend to have defined the other arms of such a balance.

Figure with letters.

Beside the figure:

a e, e f are real together with the pendant e c; the arm of resistance is potential, that is a b; the arm of the power a d is also potential. But the arm of resistance is quadruple to the arm of the power a d; wherefore one pound placed at b, the end of the arm a b, resists 4 pounds placed at the end of the arm a d moved by the pendant d f, etc.

Here, as has been said, one calculates with mathematical lines and not with the weight of the members of the instrument.

Figure with letters:

Here all the lines are potential, and the two beams n a, n m with the pendant n c are real, with the weight c.

Beside the figure:

Wishing to know what gravity the weight c, joined to the cord n c, gives of itself to that cord, and also with what gravity such weight loads itself upon the bases of the beams n a, n m, we shall have recourse to the right-angled junction created by the arms of the balance with their pendants, and with the help of such an arm we shall give true knowledge of these powers.

Here we have true notice how the weight c, which is 8 pounds, loads itself with 4 pounds upon the base of the beam n m, and with another 4 pounds upon the base of the beam n a, which are 8 pounds in all. And this is natural weight; and besides this, such bases receive a quantity of accidental weight, which weight by another name is called force, which gains so much power as the upper junction of the beams is made with a more obtuse angle.

The balance a b, a o of the weight that the base of the beam n a receives has its arms in double proportion, of which arms a o is the lever and a b its counter-lever. Hence 8 pounds, placed at the pendant o p of the sub-double arm a o, resists the descent of the double arm a b with half the weight 8, that is with 4 pounds; at the end of the double arm one resists the power of eight pounds placed on the sub-double arm a o. It follows that here arise 8 pounds of weight and 8 pounds of force, besides the natural weight of the instrument.

Figure with letters.

Beside the figure:

Four are the pendants supporting the weight m, namely those fixed in the arms of the balance, of which f g, f b are on one side, d n, d c on the other, and these are real; and of these two are semi-real, namely d n, n c; and the potential part e d is the real part of d c; o b is the potential part of f b. But of the arms of the balance there are two real and three potential: the real are a g, a b; the potential are a b, a c, a e; and this is the quality of this machine, of which, in its place, its power and validity will be fully defined.

Figure with letters.

Beside the figure:

The power of the real pendants joined to the opposite arms of the balance by an oblique position are not the true arms of the balance, being of simple denomination, that is, real position and not potential. But the true denomination is composed of real and potential, which happens when the junction of the real pendants with the arm of the balance is right-angled; and this not being right-angled, it is necessary to perform the calculation of the weights placed on the arms of such balances with the potential right angles, there being no real right angles.

Figure with letters and numbers.

Beside the figure:

Therefore, from what is declared above, we find, in the balance figured here, that the junction of the five pendants is not right-angled with the arms of its balance, that is, it is an acute junction, as the pendants e m, f m, n g, h i, K i show. Wherefore, wishing the true calculation made with the true pendants, it is necessary to use the potential pendants joined in right-angled junction to the potential arms of the said balance, which are a, b above the balance, c, d below it, which are right angles composed of the true potential arms and the true potential pendants. There remains for us a right angle below the pole of the balance, which is named a semi-real angle because it is composed of a real pendant and a potential arm.

Having given the definition and names of the arms and pendants of the balance, it is necessary to give their calculation to conclude their true powers; wherefore we shall situate the right and left potential arms around the pole of the balance by means of the circulation made above, and in the same proportions we shall divide the weight h and give to each pendant its part.

a b, b c, b f are real powers; a e is the potential arm to which the right-angled junction is made at the point e with the real pendant e f; therefore the proportion that a e has to a d, such the motive power d c has to the power that resists e f.

The lever a d stands with 4 against 3 of the counter-lever a e. Therefore the pendant d c lifts 4 at the pendant e f.

0118 v

Figura con lettere.

A lato della figura:

Figura con lettere.

A lato della figura:

Figura con lettere:

A lato della figura:

0117 r

Figura con lettere.

A lato della figura:

Figura con lettere.

A lato della figura:

Figura con lettere e numeri.

A lato della figura:

Figure with letters.

Beside the figure:

Figure with letters.

Beside the figure:

Figure with letters:

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Figure with letters.

Beside the figure:

Figure with letters.

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Figure with letters and numbers.

Beside the figure:

Balances analysed by right-angled pendants and arms

On this page

The true length of a balance arm

A fourfold resistance

Natural weight and accidental force

Naming the arms and pendants of a machine

Acute junctions must be recast as right angles