Book of Various Mathematical and Physical Speculations

I did not have the opportunity. Furthermore, although some propositions may be found in these books that have a logic separate from others, they are nonetheless not to be despised; perhaps they will open a way for someone else to progress further. For just as, for example, from the sub-contrary construction, there was later taken that divine delineation of the Planisphere which is read under Ptolemy’s name; and just as from the penultimate proposition of the first book of Euclid, which Pythagoras devised, nearly innumerable beautiful consequences in Astronomy, in Architecture, and in many other sciences have been derived; nay, just as many excellent things have been deduced from individual propositions devised by our ancestors, so perhaps it will happen that from some of my inventions, no small amount of utility may be derived in the future. If, however, you find anything here that does not please your temperament, let that sentiment of the most prudent man come to mind: As many heads, so many opinions; and it happens very rarely that the same thing can be approved by and please everyone, and it is with great difficulty that a man is found who is pleased by all the things that satisfy another. Nor should it move you that you do not see these theorems or meditations arranged in that order in which you might think they ought to be placed, both in Arithmetical matters and in the rest. For since in matters of this kind order is not necessary, it seemed to me that I could, without blame, neglect it, since I decided I must apply myself so principally to speculation or invention that I did not consider it worthwhile to expend labor and consume time in their arrangement; which same thing I did in the arrangement of the letters, in which almost no order of rank of the persons to whom I write has been observed, nor of the time in which they were written, regard having been had only for the subjects sought. Nor would I wish anyone to wonder that, in examining the properties of numbers, I use geometric figures; for so Euclid did in the second book, which method pleases me the more, the less abstract it is, because it is necessary for one who understands to contemplate mental images, since, furthermore, it is clear that every discrete quantity arises in some way from the division of the continuous, whether in act or in potency. Then, if perhaps in my demonstrations I shall seem to you at times too brief, know that the cause was that I was writing there to men practiced in these disciplines, for whom it was enough to indicate the matter. Moreover, I chose to call all the propositions of the Arithmetical volume by the name of theorems rather than problems, because the speculative part of them is mine alone, even if I have added an operative part from various propositions of that kind. Since, however, it happened in many places that for the sake of judging the truth it was necessary for me to oppose the opinions of certain men, I would not wish you to attribute this

to me as a vice, and to hold me on this account a fault-finder and slanderer because I reveal the errors of others; since rather thanks ought to be given to me, because laboring at times in those things (which Antisthenes said were more necessary in disciplines, namely, that bad things be first unlearned), I strive to uproot false opinions and show the truth, which every philosopher, following the example of Aristotle, ought to value more than the authority or favor of any man. And when you read anything of this kind in this volume,