column contains the differences between the diameters of the fly-wheel and pulley, estimated in decimal parts of the distance between their axes, which is, throughout, regarded as the unit. In the second column, are inserted the corresponding excesses of the length of the band above that of the circumference of the pulley; these excesses being, for the sake of interpolation, accompanied by their differences. And the third column exhibits the excesses of the length of the band above the circumference of the fly-wheel, with their differences. The numbers in the first and second columns go on increasing, but those in the third column decrease.

All the dimensions of any turning-lathe must be divided by the number which expresses the distance between the axes, before any of them can be sought for in this table; and the results obtained from the table must again be multiplied by the number formerly used as a divisor, in order to obtain the quantities sought for. But this calculation may be avoided, by forming a scale of the tenth, hundredth and thousandth parts of the distance between the axes, and by using this scale in all the measurements. The latter method will, in all probability, be found the most convenient. As examples of the use of the Table, I will propose two questions.

I. On the pulley of a turning-lathe are already two grooves, one of 2.4, and the other of 5.0 inches diameter. The centre of the fly-wheel is distant 30 inches from that of the pulley, and the larger groove to be made on the fly is 25 inches in diameter. Required the diameter of the other groove to be made on the wheel?

Dividing all these dimensions by 30, we obtain unit for the distance between the axes, which is the distance assumed in the table; 0.08 for the diameter of the lesser, 0.1666 for that of the greater groove on the pulley, and 0.833 for that of the greater groove on the fly-wheel.

These numbers are just what would have been found on taking the dimensions with the scale above described.

In order to find the length of the band, we take the difference between 0.8333 and 0.08, which is 0.75333, and enter,

with this number, the first column of the table. The nearest number which we can find is 0.75, opposite to which, in the second column, is found 3.32044. To correct this for the remaining figures 333, we multiply by these the tabular difference 1958, cutting off as many figures from the right of the product as there are figures in the multiplier: this done, we obtain the correction 653, which, added to 3.32044, gives 3.32697 for the correct excess of the band above the circumference of the pulley. But, if we multiply 3.1415926 by .08, the diameter of the pulley, we have .25133 for its circumference; so that the whole length of the band, the sum of 3.32697 and .25133, must be 3.57830.

In order to compute the size of the new groove to be cut in the wheel, we observe, that, as the band now passes over a pulley whose diameter is 0.16666, its excess above the circumference of that pulley, which circumference is .52360, is 3.05470. Entering the second column of the table in search of this number, and taking that which is immediately less, we find 3.05195, which has 0.61 opposite to it in the first column; to obtain the correction for this number, we divide 275, the error, by 1884, the tabular difference, affixing as many ciphers to 275 as we wish to obtain new decimal places: the result of this division is 146, whence the true difference between the diameters of the fly-wheel and pulley is 0.61146; but the diameter of the pulley is 0.16667, wherefore that of the fly-wheel is 0.77813.

These two results, multiplied by 30, give, for the length of the band, 107.349 inches; for the diameter of the new groove, 23.344 inches.

II. The distance between the axes of a turning-lathe being 32 inches; and two grooves on the fly-wheel having 38 and 34 inches for their diameters: the lesser groove on the pulley is to be 3 inches in diameter; required the size of the other?

Dividing all the dimensions by 32, we obtain, for the diameters of the wheels 1.1875 and 1.0625, and for that of the lesser groove on the pulley .09375.

Entering the first column of the table for 1.09375, which is the difference between the diameters of the first pair of grooves,

we find in the third column, opposite 1.09, the number 0.59297: To correct for the remaining figures, multiply 992, the tabular difference, by 375, and cut off three places; these operations give 372, which has to be subtracted from 0.59297, because the numbers in the third column grow less. The true excess of the band above the circumference of the wheel, is thus 0.58925. But the diameter of the wheel is 1.1875, therefore its circumference is 3.73064, and the whole length of the band 4.31989.

The circumference of the second groove on the fly-wheel is 3.33794, wherefore the excess of the band above that circumference is 0.98195. Entering the third column for this number, and taking the one immediately greater, we find, opposite to 0.98809, 0.73 in the first column. Dividing the error 614 by 1195, the tabular difference, we obtain 514, which, annexed to 0.73, gives 0.73514 for the true difference between the diameters. But the diameter of the wheel is 1.0625; wherefore that of the pulley is .32736.

These results, multiplied by 32, give,

For the length of the Band,

.138.2365 inches.

For the diameter of the New Groove, 10.4755 inches.

If it be wished to allow for the thickness of the band, we have only to add to the calculated lengths, the circumference of a circle which has the thickness of the band for its diameter: the diameters of the grooves will be in no way affected by it.

With regard to the accuracy of the calculations, it may be mentioned, that there is no probability of an error of the thousandth part of an inch in any of the diameters. To this degree of exactness few will pretend to work.