Part IV, Chapter 10: Reply to what is said by the Geometers on this subject
THERE are some geometers who think they have justified these defects by saying that they have paid no attention to them, that it is enough for them that they say nothing which they do not prove in a convincing manner, and that they are, in this way, assured of having found the truth, which is their sole aim.
It must be allowed, moreover, that these defects are not so considerable, but that we are compelled to acknowledge that of all human sciences there are none which have been better handled than those which are comprised under the general name of mathematics. All that we maintain is that something may be added to them, in order to render them more perfect, and that, though the principal thing that ought to be considered is to advance nothing but what is true, it is, nevertheless, to be desired that more attention had been paid to the more natural manner by which truth is conveyed to the mind.
For the geometers may say, if they please, that they do not care about the true order, or whether they prove by near or distant ways, provided that they accomplish what they seek, which is to convince; but they cannot change, in this way, the nature of our mind, nor prevent us from having a knowledge much more accurate, more entire and complete, of things which we know through their true causes and principles, than of those which are proved to us only through foreign and indirect ways.
It is indubitable, indeed, that we learn with incomparably greater facility, and retain much better, what has been taught us in the true order; because the ideas which have a natural connection arrange themselves much better in our memory, and suggest each other much more readily.
We may say, indeed, that what we have once known, by having penetrated into its true reason, is not retained by the memory but by the judgment, and that it becomes so thoroughly our own that we are unable to forget it; whereas what we know only by demonstrations which are not founded on natural reasons, escapes us easily, and is with difficulty recovered when it has once passed from memory, because our mind furnishes us with no means of recovering it.
It must be conceded, therefore, that it is in itself much better to observe this order than not to observe it. But all that can be said with justice is, that a small inconvenience must be neglected when we cannot avoid it without falling into a greater; that thus it is an inconvenience that the true order is not observed, but that it is better, nevertheless, to disregard it than to fail of proving invincibly that which we advance, and to expose ourselves to the danger of falling into error and paralogism, by seeking after proofs which are more natural, but which are not so convincing nor so free from all suspicion of deception.
This reply is very reasonable; and I confess that we must prefer, in all things, the certainty of not being deceived, and that the true order must be neglected if we cannot follow it without losing much of the force of the demonstrations, and exposing ourselves to error. But I do not concede that it is impossible to observe both; and I believe that a work on the elements of geometry can be made in which all things should be treated in their natural order, all propositions proved in very simple and natural ways, and in which, nevertheless, everything should be most clearly demonstrated. (This has since been accomplished in the NEW ELEMENTS OF GEOMETRY, and particularly in the new edition which has lately appeared.)