Volume 37, 1904
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### Art. IX.—On the Construction of a Table of Natural Sines by Means of a New Relation between the Leading Differences.

[Read before the Wellington Philosophical Society, 2nd November, 1904.]

#### Part II.

 1 To show the immense power of this method of obtaining the leading differences, an exceptional example is here given of the formation of a table of natural sines to twenty-four decimal places for every nine degrees of the quadrant.

The values of sin 9° and cos 9° are readily obtained from the series given in Part I.,* and are—

Sin 9° = 0.15643, 44650, 40230, 86901, 0105

Cos 9° = 0.98768, 83405, 95137, 72619, 0040

To test these values they are squared, and give:—

Sin2 9° = 0.02447, 17418, 52423, 21394, 1780

Cos2 9° = 0.97552, 82581, 47576, 78605, 8219

hence the values are correct.

Now, k = 2 (1—cos Δx) = 2 (1—cos 9°)

= 0.02462, 33188, 09724, 54761, 99195.

The leading differences are formed as described in Part I, and a convenient working schedule is arranged thus:—

Tabular interval = Δx = 9°

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

 sin 0° = 0 00000, 00000, 00000, 00000, 0000 Δ sin 0° = + 0.15643, 44650, 40230, 86901, 0105 sin 9° = sin 0° + Δ sin 0° = + 0.15643, 44650, 40230, 86901, 0105 Δ2 sin 0° = - k. sin 9° = - 0.00385, 19357, 05514, 31391, 79172 Δ sin 9° = Δ sin 0° + Δ2 sin 0° = + 0.15258, 23293, 34716, 55509, 2188 Δ3 sin 0° = - k. sin 9° = - 0.00375, 70882, 64602, 87371, 61559 Δ2 sin 9° = Δ2 sin 0° + Δ3 sin 0° = - 0.00760, 90239, 70117, 18763, 40731 Δ4 sin 0° = - k. Δ2 sin 9° = + 0.00018, 73594, 23047, 03150, 04117, 13 Δ3 sin 9° = Δ3 sin 0° + Δ4 sin 0° = - 0.00356, 97288, 41555, 84221, 57442 Δ5 sin 0° = - k. Δ3 sin 9° = + 0.00008, 78985, 71329, 89818, 89906, 78 Δ4 sin 9° = Δ4 sin 0° + Δ5 sin 0° = + 0.00027, 52579, 94376, 92968, 94023, 91 Δ6 sin 0° = - k. Δ4 sin 9° = - 0.00000, 67777, 65350, 46850, 65814, 220

[Footnote] * Trans. N.Z. Inst., 1902, p. 409–10.

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[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

 Δ5 sin 9° = Δ5 sin 0° + Δ6 sin 0° = + 0.00008, 11208, 05979, 42968, 24092, 56 Δ7 sin 0° = - k. Δ5 sin 9° = - 0.00000, 19974, 63467, 73330, 64528, 5908 Δ6 sin 9° = Δ6 sin 0° + Δ7 sin 0° = - 0.00000, 87752, 28818, 20181, 30842, 811 Δ8 sin 0° = -k. Δ6 sin 9° = + 0.00000, 02160, 75256, 81886, 56155, 999392 Δ7 sin 9° = Δ7 sin 0° + Δ8 sin 0° = - 0.00000, 17813, 88210, 91444, 08372, 5969 Δ9 sin 0° = - k. Δ7 sin 9° = + 0.00000, 00438, 63689, 84123, 11107, 46185 Δ8 sin 9° = Δ8 sin 0° + Δ9 sin 0° = + 0.00000, 02599, 38946, 66009, 67263, 45577 £10 sin 0° = - k. Δ8 sin 9° = - 0.00000, 00064, 00559, 55467, 55455, 96711, 3

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:—

 Sines Corrections. 0° 0.00000, 00000, 00000, 00000, 0000 9° 0.15643, 44650, 40230, 86901, 0105 18° 0.30901, 69943, 74947, 42410, 2293 27° 0.45399, 04997, 39546, 79156, 0408 36° 0.58778, 52522, 92473, 12916, 8706 45° 0.70710, 67811, 86547, 52440, 0845 −1 54° 0.80901, 69943, 74947, 42410, 2295 −2 63° 0.89100, 65241, 88367, 86235, 9712 −3 72° 0.95105, 65162, 95153, 57211, 6443 −4 81° 0.98768, 83405, 95137, 72619, 0045 −5 90° 1.00000, 00000, 00000, 00000, 0006 −6

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°.

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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:—

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

 Tabular interval = Δ x = 1° k = 0.00030, 46096, 87217, 52168, 59768, 818 sin 0° = 0.00000, 00000, 00 Δ sin 0° = + 0.01745, 24064, 373 sin 1° = sin 0° + Δ sin 0° = + 0.01745, 24064, 37 Δ2 0° = - k. sin 1° = - 0.00000, 53161, 721 Δ sin 1° = Δ sin 0° + Δ2 sin 0° = + 0.01744, 70902, 652 Δ3 sin 0° = - k. Δ sin 1° = - 0.00000, 53145, 5271 Δ3 sin 1° = Δ2 sin 0° + Δ8 sin 0° = - 0.00001, 06307, 248 Δ4 sin 0° = - k. Δ2 sin 1° = + 0.00000, 00032, 38222 Δ8 sin 1° = Δ3 sin 0° + Δ4 sin 0° = - 0.00000, 53113, 1449 Δ5 sin 0° = - k. Δ3 sin 1° = + 0.00000, 00016, 17877, 8 &c.

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:—

[Footnote] * See the value of sin 1° in Logarithmic Tables, R. Shortrode— sin 1° = 0.01745, 24064, 17275, 54
which is 0.00000, 00000, 20007, 97 in error.

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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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 sin 9° = + 0.15643, 44650, 40 Δ sin 9° = + 0.01721, 37126, 267 sin 10° sin 9° + Δ sin 9° = 0.17364, 81776, 67 Δ2 sin 9° = - k sin 10° = −0.00005, 28949, 1709 Δ sin 10° = ° sin 9° + Δ2 sin 9° = + 0.01716, 08177, 096 Δ3 sin 9° = - k sin 10° = - 0.00000, 52273, 51315 Δ2 sin 10° = - 0.00005, 81222, 6840 Δ4 sin 9° =−k Δ2 sin 10° = + 0.00000, 00177, 04606, 0 Δ3 sin 10° = - 0.00000, 52096, 46709 Δ5 sin 9° = + 0 00000, 00015, 86908, 85 Δ4 sin 10° = + 0.00000, 00192, 91514, 9 Δ6 sin 9° = - 0.00000, 00000, 05876, 382 &c.

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:—

 Sines. Sines. 9° 0.15643, 44650, 40 14° 0.24192, 18956, 00 10° 0.17364, 81776, 67 15° 0.25881, 90451, 03 11° 0.19080, 89953, 77 16° 0.27563, 73558, 17 12° 0.20791, 16908, 18 17° 0.29237, 17047, 23 13° 0.22495, 10543, 44 18° 0.30901, 69943, 75

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

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

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

= 0.00086, 18610, 01

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Section 9° 0′–00°.

 sin 9° = + 0.15643, 44650, 4 Δ sin 9° = + 0.00086, 18610, 01 sin 9° 3′ = + 0.15729, 63260, 4 Δ2 sin 9° = - k. sin 9° 3′ = - 0.00000, 01197, 880 Δ sin 9° 3′ = Δ sin 9° + Δ2 sin 9° = + 0 00086, 17412, 13 Δ3 sin 9° = - k. Δ sin 9° 3′ = - 0.00000, 00006, 5625 &c.

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:—

° Sines.
9 0 0.15643, 44650, 4
3 0.15729, 63260, 4
6 0.15815, 80672, 5
54 0.17192, 91003, 8
57 0.17278, 87047, 7
10 0 0.17364, 81776, 7
10 0 0.17364, 81776, 67 check value

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

 Tabular interval = Δx = 10″. Initial value, 9° 54′. sin 10″ = 0.00004, 84813, 68092, 4 cos 10″ = 0.99999, 99988, 24778, 473 k = 0 00000, 00023, 50443, 053 Δ sin 9° 54′ = cos 9° 54′. sin 10″ - k/2 sin 9° 54′ = 0.00004, 77592, 458

Section 9° 54′–9° 57′.

 sin 9° 54′ = + 0 17192, 91003 Δ sin 9° 54′ = + 0.00004, 77592, 458 sin 9° 54′ 10″ = + 0 17197, 68595 Δ2 sin 9° 54′ = - k. sin 9° 54′ 10″ = - 0.00000, 00004, 04221, 8 Δ sin 9° 54′ 10″ = + 0.00004, 77588, 416 Δ3 sin 9° 54′ = - k. Δ sin 9° 54′ 10″ = - 0 00000, 00000, 00112, 254 &c.

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,

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while the value obtained in 3 above is 0.17278, 87047, 7, which sufficiently checks the work.

The results may then be safely cut down to seven decimals, and will give the sines correct in the last figure, except perhaps in a few cases where the 8th, 9th, and 10th decimals are 499, 500, or 501

Thus in the section under construction the final values are:—

° Sines. Proportional Parts
9 54 0 0.171 9291
10 171 9769
20 172 0246
30 172 0724
40 172 1201
50 172 1679 1 47.8
55 0 0.172 2156 2 95.5
10 172 2634 3 143.3
20 172 3112 4 191.0
30 172 3589 5 238.8
40 172 4067 6 286.5
50 172 4544 7 334.3
56 0 0.172 5022 8 382.0
10 172 5499 9 429 8
20 172 5977
30 172 6454
40 172 6932
50 172 7410
57 0 0.172 7887

The mark under a figure, thus 2, in sin 9° 56′ 40″, indicates that the figure so marked has been increased by 1 to compensate for the figures cut off, the full value in this instance being 0.17269319964.