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The construction of the wonderful canon of logarithms

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-<br />

In<br />

.<br />

Notes on Trigonometrical Propositions. 77<br />

logarithm add, as before, 693147, and <strong>the</strong> sum, that is<br />

<strong>the</strong> number remaining since <strong>the</strong> sines are contrary, will<br />

be 638826, half <strong>of</strong> which, 319413, is <strong>the</strong> logarithm <strong>of</strong><br />

46° 36' o". <strong>The</strong> arc <strong>of</strong> <strong>the</strong> given logarithm is <strong>the</strong>refore<br />

93° 1 2' o", <strong>the</strong> versed sine <strong>of</strong> which is 105582 16, and since<br />

this is greater than radius it has a negative logarithm,<br />

namely —54321.<br />

Demonstration.<br />

, j-versed sine <strong>of</strong> arc] ,<br />

X c ") cont. X a "j cont. ix c, sine <strong>of</strong> 30° o' \ cont.<br />

c g > pro- a e > pro^ c g, sine <strong>of</strong> ^ arc c d > proc<br />

h ) port. a f )<br />

port. c b, double <strong>of</strong> line c h ) port.<br />

.Letter on I observed that <strong>the</strong> sixth proposition might be<br />

proved in an exactly similar way.<br />

Of <strong>the</strong> spherical triangle A B D ]<br />

finding <strong>the</strong> base we may pursue ano<strong>the</strong>r method^<br />

namely:—<br />

Add <strong>the</strong> logarithm <strong>of</strong> <strong>the</strong> versed sine <strong>of</strong> <strong>the</strong> given angle to <strong>the</strong><br />

<strong>logarithms</strong> <strong>of</strong> <strong>the</strong> given sides, and <strong>the</strong> sum will be <strong>the</strong> logarithm <strong>of</strong><br />

<strong>the</strong> difference between <strong>the</strong> versed sine <strong>of</strong> <strong>the</strong> difference <strong>of</strong> <strong>the</strong> sides<br />

and <strong>the</strong> versed sine <strong>of</strong> th£ base required. This difference being<br />

consequently known, add to it <strong>the</strong> versed sine <strong>of</strong> <strong>the</strong> difference <strong>of</strong> <strong>the</strong><br />

sidesy and <strong>the</strong> sum. will be <strong>the</strong> versed sine <strong>of</strong> <strong>the</strong> base required.<br />

For example, let <strong>the</strong> sides be .34° and 47°, <strong>the</strong>ir loga-<br />

K 3 rithms

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