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309017, &c. (equal to Half the Side of a Decagon infcribed in the Circle) let the Co-fine of 3°, the Difference between 18° and 15°, be found *; from which the Co-fine of 45′ will be had, by two Bifections only: Whence the Sines of all the Arches in the Progreffion 1° 30', 2° 15', 3° 00', 3° 45', &c. may be determined (by Theor. 1. p. 13.) and that to any affigned Degree of Exactnefs.

The Sines of all the Terms of the Progreffion 45′, 1°30′, 2° 15', &c. up to 60°, being thus derived, the next thing is to find, by Help of thefe, the Sines of all the intermediate Arches, to every fingle Minute.

This, if you defire no more than the 4 or 5 first Places of each (which is exact enough where nothing less than Degrees and Minutes is regarded), may be effected by barely taking the proportional Parts of the Differences.

But if a greater Degree of Accuracy be infifted on, and you would have a Table carried on to 7 or 8 Places, each Number (which is fufficient to give the Value of an Angle to Seconds, and even to Thirds, in most Cases), then the Opperation may be as follows:

1. Multiply the Sum of the Sines of any two adjacent Terms of the Progreffion 45', 1° 30′, 2° 15′, 3° 00, 3° 45', &c. (betwixt which you would find all the intermediate Sines) by the Fraction 0000000423, for a first Product; and this, again, by 22, for a fecond Product; to which laft, let of the Difference of the two propofed Sines (or Extremes) be added, and the Sum will be the Excefs of the firft of the intermediate Sines above the leffer Extreme.

*Note, The Co-fine of the Difference of 2 Arches (fuppofing Radius Unity), is found by adding the Product of their Sines to that of their Co-fines; as is hereafter demonftrated.

2o. From

2o. From this Excefs let the firft Product be continually fubtracted; that is, first, from the Excefs itself; then from the Remainder; then from the last Remainder, and fo on 44 Times.

3°. To the leffer Extreme add the foremention'd Excefs; and, to the Sum, add the first Remainder ; to this Sum add the next Remainder, and fo on continually: Then the feveral Sums thus arifing will refpectively exhibit the Sines of all the intermediate Arches, to every fingle Minute, exclufive of the laft; which, if the Work be right, will agree with the greater Extreme itself, and therefore will be of Ufe in proving the Operation.

But to illuftrate the Matter more clearly, let it be propofed to find the Sines of all the intermediate Arches between 3° 00′ and 3° 45', to every fingle Minute, thofe of the Extremes being given, from the foregoing Method, equal to 05233595 and 06540312 refpectively. Here, the Sum of the Sines of the Extremes being multiplied by ,0000000423, the firft Product will be ,00000000498, &c. or,0000000050,nearly, (which is fufficiently exact for the prefent Purpose); and this, again, multiplied by 22, gives ,0000001 I for a 2d Product; which added to,0002903815, Part of the Difference of the two given Extremes, will be ,0002904915, the Excefs of the Sine of 3° or' above that of 3° 00'. From whence, by proceeding according to the 2d and 3d Rules, the Sines of all the other intermediate Arches are had, by Addition and Subtraction only. See the Operation.

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45

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,000290 4915 Excefs

,0000000050

,05233595 Sine 3° 0′ ,0002904915

4865 1 Rem. 0526264415 Sine 3° 1′

50

2904865

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4665 5th Rem. 0537883575 Sine 3° 5′

50

2904665

46156th Rem. 0540788240 Sine 3o 6′

50

2904615

4565 7th Rem. 0543692855 Sine 3° 7′

50

2904565

45158th Rem. 0546597420 Sine 3° 8′

50

2904515

4465 9th Rem. 0549501935 Sine 3° 9′

3°9′

50

2904465

4415 10th Rem.
&c.

,0552406400 Sine 3° 10° &c.

Again, as a fecond Example, let it be required to find the Sines of all the Arches, to every Minute, between 59° 15′ and 60° 00'; thofe of the two Extremes being first found, by the preceding

Method.

Method. In this Cafe, the two Extremes, being ,85940641 and ,86602540, their Sum will be = 1,72543, &c. and their Difference = ,00661899; whereof the former, multiplied by ,0000000423 (fee the Rule) gives ,00000007298, &c. or ,0000000730, nearly, for the first Product (which is exact enough for our Purpose); therefore the 2d Product, or ,0000000730 x 22, will be ,0000016060; which, added to of the Difference, gives ,0001486947; from whence the Operation will be as follows:

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84757 3 Rem. ,8598522751 Sine 59° 18′

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84027 4th Rem. ,8600007508 Sine 59° 19′

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83297 5th Rem. 8601491535 Sine 59°20′

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82567 6th Rem. ,8602974832 Sine 59° 21′

&c.

1482567

,8604457399 Sine 59° 22′′

&c.

After the fame Manner the Sines of all the intermediate Arches between any other two propofed Extremes may be derived, even up to 90 Degrees;

C 2

but

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but thofe of above 60° are beft found from those below, as has been fhewn elsewhere.

The Realons upon which the foregoing Operations are founded, depend upon Principles too foreign from the main Design of this Treatise, to be explained here (even would Room permit); however, as to the Correctness and Utility of the Method itself, I will venture to affirm, that, Whoever has the Inclination, either to calculate new Tables, or to examine those already extant, will not find one Quarter of the Trouble, this Way, as he unavoidably must according to the common Methods.

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