8 CONSTRUCTION OF THE CANON.
2.Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical, one which advances by unequal and proportionally increasing or decreasing intervals.
Arithmetical progressions: 1, 2, 3, 4, 5, 6 7; &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c. Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.
3.In these progressions we require accuracy and ease in working. Accuracy ts obtained by taking large numbers for a basis; but large numbers are most easily made from small by adding cyphers.
Thus instead of 100000, which the less experienced make the greatest sine, the more learned put 10000000, whereby the difference of all sines is better expressed. Wherefore also we use the same for radius and for the greatest of our geometrical proportionals.
4.In computing tables, these large numbers may again be made still larger by placing a period after the number and adding cyphers.
Thus in commencing to compute, instead of 10000000 we put 10000000.0000000, lest the most minute error should become very large by frequent multiplication.
5.In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which ts unity with as many cyphers after it as there are figures after the period.
Thus 10000000.04 is the same as also 25.803 is the same as ; also 9999998
0005021
