The New Thoughts of Galileo

fourth, as eighteen to eight: if the fifth, as twenty-one to nine, and so on to infinity.

Finally, to find their tangents, in the first of these curves, if it is touched at point C, by the straight line C E. B E will be double of B D, and triple of the same B D in the second, quadruple in the third, and quintuple in the fourth, and so on to infinity.

I come now to the aliquot parts original: "parties aliquotes", meaning factors or divisors, which are more difficult to find than any other difficulty in Geometry: from this comes that many have not been able to master them. Now the first number from which one has taken the subject to work on it, is 120, whose aliquot parts make the double, to wit 240. One had never found others that I know of, and even most Analysts did not know if there were any like them, until excellent Geometers, Analysts, and Arithmeticians have added lately 672, 523776, and 1476304896, which have the same property; and besides, an excellent mind has found that the number which follows, whose aliquot parts make also the double, to wit 459818240, being multiplied by 3, that is to say being tripled, produces the number 1379454720, whose aliquot parts make the triple. They have also found ones which are sub-triples of their aliquot parts, for example, those which follow, 30240, 32760, 23569920, 45532800, 142990848, 43861478400, 66433720320, 403031236608, to which they can add a thousand others which will have the same property, and even which will be quadruples of their aliquot parts, as are the three which follow, 14182439040, 50866680320, and 30823866178560, and as many as one would want of others, whose aliquot parts will be the quintuple, the sextuple, the centuple, etc., to infinity: which had not been known until now. One had also not known other numbers, whose aliquot parts taken alternatively reproduced the same amicable numbers original: "nombres amiables", pairs of numbers where each is the sum of the aliquot divisors of the other, than 284 and 220, which are called amicable, because the aliquot parts of 284 make 220, and those of 220 make 284. But one has recently found the two couples which