Euclid's Elements

[He] gathered what was dispersed, arranged what was gathered, and himself reduced to absolute and anelenktous irrefutable demonstrations those things which had been demonstrated more crudely or negligently. Great is the praise of his predecessors, indeed, but much greater is that of Euclid, who composed them in such an ordered manner that, by this one feat alone, he secured for himself eternal praise among men of sound mind. He completed what was begun and rendered the uncertain most certain by the firmest of reasons, so that almost nothing more can be desired in it. Now, nearly two thousand years have passed since Euclid was counted among the living. He has had many adversaries who, out of a disease of envy rather than a love of truth, have attempted to undermine his writings with all their effort; yet, stern investigators have still not been able to detect any pseudographian false writing/error, no mistake, or any paralogism in them.

The other monuments of this most excellent man are considered to be these: he wrote the Optics, Catoptrics, Music, Data, and Phenomena. He also wrote a book on divisions, four books on conics, and three on porisms, as is evident from Proclus and Pappus, although these have not reached our hands. These are the things we have been able to find regarding our Euclid, through whose immortal benefit Mathematics—which, having crossed the Greek sea from Egypt, had lived in Greece for two hundred years and a little more—has attained its dignity and its honors, not without the will of the gods.

Now, let the opinion that disturbs the minds of students not a little regarding the demonstrations of the Elements be recounted briefly. Although this dispute brings no utility to the future geometer, it keeps the lovers of this discipline quite solicitous, I know not how, because they desire to know to whom they should attribute such a great benefit and such a singular gift. Among others who have disputed this matter, Ioannes Buteo and Petrus Ramus, men of very sharp judgment, have gone into completely opposite opinions. The latter, in his preface to Mathematics, thinks not only that the demonstrations should be ascribed to Theon (which others have also said), but even the Elements themselves; this is because [Euclid] was the final stoicheiotes element-writer, and the invention of no proposition is attributed by Proclus to Euclid among his praises, and also because Theon himself specifically praised his own editions of the Elements in his first commentary on Ptolemy's Great Construction, such that Theon can claim the Elements for himself by the same right that Euclid did before. He also proves this with the reason that the demonstrations of Euclid, which are read in the commentaries of Proclus, do not at all agree with those we have in the Elements.

The former (I speak of Buteo), in his annotations on Euclid, explicitly denies this; he defends the ancient praise of the most distinguished man. This is because, among the ancients, theorems were never brought forward without demonstration, as they would have no utility or dignity if they were naked. Moreover, it is probable that those words ek ton Theonos synousion from the discussions of Theon, from which the whole occasion for this dispute flowed, could be understood to mean that we say Theon indeed wrote commentaries on the Elements, but they perished through the calamity of time—just as those which Pappus of Alexandria wrote on the same—while the title was preserved, which was later carelessly added to Euclid himself.

We, however, following a middle path, believe that the books on the Elements were left to us by Euclid, adorned with his own demonstrations. For how can we doubt this, when Proclus, in his commentaries on the tenth proposition, after reciting the demonstration of Apollonius of Perga, adds these words: pollo oun kreitton he tou stoicheiotou apodeixis therefore, the proof of the element-writer is by far better—for this is how he calls Euclid—and simpler, and more based on principles. But as we affirm this truly, so we will deservedly concede that Theon, a man of excellent talent, brought to light the demonstrations of Euclid explained more diffusely and more clearly, which can be observed in Proclus. Thus, the Data are not held in exactly the same way as in the seventh book of the mathematical collections of Pappus. Nor are the Optics or Catoptrics, which we saw in Rome in the Vatican Library. Therefore, if we concede these to Euclid by the consensus of all, the Elements must also be conceded, especially since Theon differs from him more in the wording than in the essence of the method of demonstration. Those demonstrations, therefore, are indeed Euclid's, but written in the manner in which Theon, once having followed Euclid, explained them to his disciples.

It would not be useless, nor unpleasant, for the readers if I were to add the notable opinions of Plato, Xenocrates, and our own Euclid to this dispute as a concluding flourish. For they will be able to serve as a copious and elegant speech for candidates of Geometry. Therefore, Plato, to show openly the absolute necessity of the knowledge of this faculty for a future philosopher, placed these words before the door of his gymnasium: oudeis ageometretos eisito let no one unskilled in geometry enter here. Xenocrates, however, who taught third in the Academy after his master, said to someone ignorant of Mathematics and Geometry who was entering the gymnasium: "Go away, for you do not have the labas handles/grasps of philosophy." But what are we to say about our own Geometer? To Ptolemy the King, when he first asked whether there was any other...