GALILEO'S proportional compasses are said to date from the year 1597. We infer that the instrument consisted of two arms, jointed, as in the accompanying figure, so that one arm could move freely about the joint. Each arm had a number of equal divisions (not necessarily of the same length on each arm), the zero point being at the joint. To divide a given length into five equal parts it is necessary to take an ordinary pair of compasses and measure the given length with these, then set the proportional compasses so that the fifth division on each arm may be at the given distance apart, then transfer with the ordinary compasses the distance between the unit divisions—this will be one-fifth of the given line. This seems to have been the manner of using the instrument employed by Galileo (cf. Marie, Histoire des Sciences Mathématiques et Physiques, tome iii. p. 108). Other modes of using will doubtless occur to most of our readers. The principle involved in this and similar instruments, and certainly in the one before us, is that of the proportionality of corresponding sides in similar triangles.