TRACT 19. TRIGONOMETRICAL TABLES, &C. 295
cants, are continued downwards on the left-hand pages, asfar as to 45 degrees, and then returned upwards on the right-hand pages, so that the complements are always on the sameline in the opposite or facing pages.
The mathematical works of Christopher Clavius (a Ger-man jesuit, who was born at Bamberg in 1537) in five largefolio volumes, were printed at Moguntia, or Mentz, in 1612,the year in which the author died, at the age of 75. In thefirst volume we find a very ample and circumstantial treatiseon trigonometry, with Regiomontanus ’s canon- of sines, forevery minute, as also canons of tangents and secants, each ina separate table, to the radius 10000000, and in a form con-tinued forwards all the way up to 90 degrees. The expla-nation of the construction of the tables is very complete, andis chiefly extracted from Ptolemy, Purbach, and Regiomon-tanus . The sines have the differences set down for eachsecond, that is, the quotients arising from the differences ofthe sines divided by 60.
About the year 1600, Ludolph van Collen , or a Ceulen, arespectable Dutch mathematician, wrote his book “ de cir-culo et adscripts,” in .which he treats fully and ably of theproperties of lines drawn in and about the circle, and especi-ally of chords or subtenses, with the construction of the canonof sines. The geometrical properties from which these linesare computed, are the same as those used by former writers;but bis mode of computing and expressing them is differentfrom theirs; for they actually extracted all the roots, &c, atevery step, or single operation, in decimal numbers; but heretained the radical expressions to the last, making them how-ever always as simple as possible: thus, for instance, he de-termines the sides of the po-lygons of 4, 8, 16, 32, &c,sides, inscribed in the circlewhose radius is 1, to be asin the table here annexed:where the point before anyfigure, as V.2 signifies the
No. ofSides.
Length of each side.
4
^2
8
V.2-^/2
16
32
v /.2-v'- 2 + V' -2— y'2
&c.
& c.