Combinations of Pythagorean Triangles. 
507 
half angles 90° within a second; tenths of seconds can be ob¬ 
tained with six figure logarithms as a usual thing. 
After the tangents of the half angles have been computed as 
whole numbers or vulgar fractions, I need not say that they can 
readily be checked by the ordinary trigonometric, or more 
properly goniometric, formulae, which will show that the sums 
of the half angles are 90° in each case. Thus in our example 
tan 34 0 = = H 
The writer regards it as an orderly method of teaching 
Trigonometry to deal with the functions without logarithms be¬ 
fore the pupils are required to learn the trigonometric arti¬ 
ficialities. In the method which the writer prefers Trigonometry 
becomes the first mathematical subject of Freshman year, and 
the extension of Algebra is deferred till Trigonometry is pretty 
well understood, at least in its elements. 
I must defer till another occasion some suggestions relating 
to the construction of tables of the natural trigonometric 
functions. 
Table I.— Sides of Pythagorean Triangles. 
X 
y 
z 
X 
y 
z 
3 
4 
5 
33 
56 
65 
5 
12 
13 
16 
63 
65 
8 
15 
17 
48 
55 
73 
7 
24 
25 
36 
77 
85 
20 
21 
29 
13 
84 
85 
12 
35 
37 
39 
80 
89 
9 
40 
41 
65 
72 
97 
28 
45 
53 
15 
112 
113 
11 
60 
61 
17 
144 
145 
33 
