Footnote 77
Supp. Prop., §42b. Here, `by the craft of Umbra
Recta' signifies, by a method similar to that in the last article,
for which purpose the numbers must be adapted for computation by the umbra
recta. Moreover, it is clear, from fig. 17, that the numbers
4 and 3 (in lines 2 and 4) must be transposed. If the side parallel
to bE be called nm, and mn, Ec be produced
to meet in o, then mo : mE : : bE
: bc; or mo : 12 : : 12 : bc;
or mo = 144, divided by bc (= 3) = 48. Similarly, m'o'
= 144, divided by b'c' (= 4) = 36. And, as in the lost article,
the difference of these is to 12, as the space EE' is to the altitude.
This is nothing but Art. 42 in a rather clumsier shape.
Hence is appears that there are here but 3 independent propositions,
viz. those in articles 41, 42, and 43, correspoding
to figs. 16, 17, and 18 respectively. Arts. 41a
and 41b are mere repetitions of 41; 42a
and 42b, of 42; and 43a, of 43.