The construction of the wonderful canon of logarithms
The construction of the wonderful canon of logarithms
The construction of the wonderful canon of logarithms
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
-<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