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This exhibits the complete working necessary to obtain the leading differences up to Δ¹⁰ sin 0°. As will be seen from the schedule above, the only operations are addition and multiplication. Each multiplication was done to the full extent as shown above on the Brunsviga calculating-machine without any intermediate record, and each multiplication was checked by doing it in duplicate: thus, to obtain k. sin 9°, k was first set on the machine and multiplied by sin 9°, then sin 9° was set on the machine and multiplied by k, and no result was accepted unless every figure to the last agreed in each case.

Having now obtained the leading differences and the initial term (sin 0°), the table is formed in the usual way, with the following results:—

The last value is 6 in excess in the twenty-fourth decimal place; comparing the value of sin 81° with cos 9° as found direct from the series it is seen that this value is 5 in excess; and comparing sin 54° with sin 18° the former is 2 in excess: hence it seems reasonable to adjust these values by deducting 1, 2, 3, 4, 5, and 6 from sin 45°, sin 54°, sin 63°, sin 72°, sin 81°, and sin 90°.

2. As the tabular interval becomes smaller the value of k is reduced, with a corresponding reduction in the labour of forming the leading differences. Thus in the formation of a table of sines for every degree we have, from the series in Part I.,—

sin 1° = 0.01745, 24064, 37283, 51281, 94189, 78

cos 1° = 0.99984, 76951, 56391, 23915, 70115, 59

The squares of these are:—

sin² 1° = 0.00030, 45864, 90452, 13499, 68782, 7997

cos³ 1° = 0.99969, 54135, 09547, 86500, 31217, 20

hence sin 1° and cos 1° are correct,^* and k = 2 (1—cos 1°) = 0.00030, &c.

The working schedule for the formation of the leading differences is accordingly as follows:—

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With these leading differences the table may be constructed up to 60°, or to 90° if preferred; however, unless the work is carried out to considerably more decimal places than are here shown the final figures of the higher values will not be accurate.

It seems preferable, therefore, to construct the table in sections, 0°–9°, 9°–18°, 18°–27°, &c.

The values of sin 9°, sin 18°, &c., are obtained from 1 above, and the formation of the table for the section 9°–18° will then proceed as follows:—

Tabular interval = Δ x = 1°. Initial value sin 90°.

k = 0.00030, &c.

Now, Δ sin 9° = cos 9° sin 1°–k/2. sin 9°

= 0.01721, &c.

Section 9°–18°.

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These leading differences are sufficient to determine the sines to twelve places of decimals, the value obtained for sin 18° being 0.30901, 69943, 75, which checks the work; and the other values, as taken from the working schedule without any alteration or adjustment, are:—

3. The next example selected is where the tabular interval is 3′, and values approximately correct to eleven decimal places are required.

Tabular interval = Δx = 3′. Initial value 9°.

sin 3′ = 0.00087, 26645, 15235, 14954, 3304 From series in Part I.

cos 3′ = 0.99999, 96192, 28249, 43113, 77097 From series in Part I.

∴ k = 0.00000, 07615, 43501, 13772, 45806

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also Δ sin 9° = cos 9° sin 3′–k/2 sin 9°

= 0.00086, 18610, 01

Section 9° 0′–00°.

Sections of 1° each (0°–1°; 1°–2°; &c.) are formed thus, and the values found in 2 above are used to check each terminal value.

Thus for the section 9° 0′–10° the sines are:—

4. Finally we have the case of a 10″ table to seven decimals.

Section 9° 54′–9° 57′.

These leading differences are sufficient to determine the sines to ten decimal places approximately; and for this particular section the value of sin 9° 57′ is 0.17278, 87047, 6,