Page:Philosophical Transactions - Volume 003.djvu/22

This page has been validated.

(648)

By anyone of which three Series, it is not hard to calculate, as near as you please, these and the like Hyperbolic spaces, whatever be the Rational Proportion of AE to BC. As for Example, when AE is to BC, as 5 to 4. (whereof the Calculation follows after that where the Proportion is, as 2 to 1. and both by the third Series.)

First then when (in Fig.1.) AE. BC:: 2. 1.

2 x 3 x 4)	1. (0.0416666666——	] 0.0416666666
4 x 5 x 6)	1. (0.0083333333——	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ 0.0113095237
6 x 7 x 8)	1. (0.0029761904——
8 x 9 x 10)	1. (0.0013888888——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0029019489
10 x 11 x 12)	1. (0.0007575757——
12 x 13 x 14)	1. (0.0004578754——
14 x 15 x 16)	1. (0.0002976190——
16 x 17 x 18)	1. (0.0002042484——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0007306482
18 x 19 x 20)	1. (0.0001461988——
20 x 21 x 22)	1. (0.0001082251——
22 x 23 x 24)	1. (0.0000823452——
24 x 25 x 26)	1. (0.0000641026——
26 x 27 x 28)	1. (0.0000508751——
28 x 29 x 30)	1. (0.0000410509——
30 x 31 x 32)	1. (0.0000336021——
32 x 33 x 34)	1. (0.0000278520——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0001829939
34 x 35 x 36)	1. (0.0000233426——
36 x 37 x 38)	1. (0.0000197566——
38 x 39 x 40)	1. (0.0000168691——
40 x 41 x 42)	1. (0.0000145180——
42 x 43 x 44)	1. (0.0000025843——
44 x 45 x 46)	1. (0.0000109793——
46 x 47 x 48)	1. (0.0000096361——
48 x 49 x 50)	1. (0.0000085034——
50 x 51 x 52)	1. (0.0000075415——
52 x 53 x 54)	1. (0.0000067193——
54 x 55 x 56)	1. (0.0000060125——
56 x 57 x 58)	1. (0.0000054014——
58 x 59 x 60)	1. (0.0000048704——
60 x 61 x 62)	1. (0.0000044068——
62 x 63 x 64)	1. (0.0000040002——

	0.0416666666
	0.0113095237
	0.0029019589
	0.0007306482
3)	0.0001829939	( 0.0000609980
	0.05679179
+	0.00006100
	0.05685279	< EdCy
But	0.0007306482	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ ${\overset {\cdot \cdot }{\underset {\cdot \cdot }{-}}}$
	0.0001829939
	0.0000458315
Therefore	0.05679179
+	0.00004583
+	0.00001528
	0.05685290	> EdCy.

For, it has been demonstrated that 1/4 of any terme in the left Column is less than the terme next after it; and therefore that, of the last terme, at which you

stop