Part IV, Chapter 9: Of some Defects which are commonly to be met with in the Method of the Geometers
WE have seen what of excellence the method of the geometers possesses. We have reduced this method to five rules, which we cannot too thoroughly fix in our minds: and it must be confessed that there is nothing more admirable than the discovery of so many hidden things, and their demonstration, by reasons so strong and so invincible, through the employment of so few rules: so that, among all philosophers, to them alone belongs the advantage of having banished from their schools and books controversy and dispute.
Nevertheless, if we would judge of things without prejudice, as we cannot take away from them the glory of having followed a much more certain course for the discovery of truth than any others, so neither can we deny that they have not fallen into some defects, which, though they have not turned them aside from their end, have nevertheless prevented them from reaching it by the shortest and most convenient route. I will endeavour to show this by selecting from Euclid some examples of these defects.
1st DEFECT.
Paying more attention to certainty than to evidence, and to the conviction of the mind than to its enlightenment.
The geometers are worthy of all praise in seeking to advance only what is convincing; but it would appear that they have not sufficiently observed, that it does not suffice for the establishment of a perfect knowledge of any thing, to be convinced that it is true, unless, beyond this, we penetrate into the reasons, derived from the nature of the thing itself, why it is true. For until we arrive at this point, our mind is not fully satisfied, and still seeks greater knowledge than this, which marks that it has not yet a true knowledge. We may say that this defect is the source of all the others which we shall notice; and thus it is not necessary to explain it further here, since we shall speak of it sufficiently in what follows.
Second DEFECT. Proving things which have no need of proof.
The geometers maintain that it is not necessary to undertake the proof of what is clear of itself. They nevertheless often do it, because, being more bent on convincing the mind than enlightening it, as we have said, they believe that they shall convince it better by finding some proof of those things even which are most evident, than by simply proposing them, and leaving the mind to recognise their evidence.
It is this which led Euclid to prove that the two sides of a triangle are greater than a single one, although this was evident from the very notion of a right line, which is the shortest possible distance between two points, and the natural measure of the distance from one point to another, which it would not be if it were not also the shortest of all lines which could be drawn from one point to another.
This is what led him, again, to make the following:—To draw a line equal to a given line—not a postulate, but a problem which must be demonstrated, although it is easy, and indeed more so than to draw a circle having a given radius.
This defect has arisen, no doubt, from its not having been sufficiently considered that all the certainty and evidence of our knowledge in the natural sciences spring from this principle:—That we may affirm of a thing all that is contained in its clear and distinct idea. Whence it follows, that when we need only, in order to recognise that an attribute is contained in an idea, to consider the idea simply, without connecting it with others, it ought to be considered as clear and evident, as we have already said above.
I know, indeed, that there are some attributes which may be seen more easily in ideas than others, but I believe that it is enough that they are able to be seen clearly with a moderate attention, and that no man, with a rightly constituted mind, is able seriously to doubt them; for those propositions, which are derived thus from the simple consideration of ideas, to be regarded as principles which have no need of proof, but, at most, of a little explanation. Thus I maintain that we cannot pay much attention to the idea of a right line without perceiving not only that its position depends on two points alone (which Euclid has taken as one of his postulates), but that we can also comprehend, without trouble, and very clearly, that if a right line cut another, and there are two points in the cutting line, each of which is equally distant from two points in the line which is cut, there will be no other point of the cutting line which is not equally distant from these two points of the line which is cut. Hence, it will be easy to determine when a line is perpendicular to another without employing either angle or triangle, which ought not to be treated of, before the things which are demonstrated by perpendiculars alone, are well established.
It is also to be remarked that there are some excellent geometers who employ, as principles, propositions less clear than these: as Archimedes, who established his beautiful demonstration on this axiom—That if two lines in the same plane have their extremities common, and are bent or hollow towards the same part, what is contained will be less than that which contains.
It must be confessed that this defect of proving what needs no proof does appear a very great one, and is not so in itself; but it is important in its consequences, since it is from this that the inversion of the natural order, of which we shall speak below, springs;—this desire of proving what ought to be assumed as clear and evident of itself, having often obliged the geometers to treat of things (in order to employ them, as proofs, in what ought not to have been proved) which, according to the order of nature, should have been treated of afterwards.
3d DEFECT.
Demonstrating by Impossibility.
Those kind of demonstrations which show that a thing is such, not by its principles, but by some absurdity which would follow, if it were not so, are very common in Euclid. It is clear, however, that while they may convince the mind, they do not enlighten it, which ought to be the chief result of knowledge; for our mind is not satisfied unless it knows not only that a thing is, but why it is, which cannot be learnt from a demonstration which reduces it to the impossible.
Not that these demonstrations are to be altogether rejected, for we may oftentimes employ them to prove negatives, which are properly only corollaries from other propositions, either clear of themselves or demonstrated before in another way; and then that kind of demonstration, by reducing them to the impossible, occupies the place rather of an explanation than a new demonstration.
We may say, in fine, that these demonstrations are allowable only when we are unable to furnish others, and that it is a fault to employ them in proving what may be positively proved. Now there are many propositions in Euclid which he has only proved in this way which may be otherwise proved without any great difficulty.
4th DEFECT.
Far-fetched Demonstrations.
This defect is very common among the geometers—they take no trouble as to the quarter whence the proofs which they furnish are obtained, provided they are convincing—to prove things by foreign methods is, however, to prove them but very imperfectly.
This will be better understood by some examples:—Euclid I., prop. 5, proves that an isosceles triangle has the two angles at its base equal, by prolonging equally the sides of the triangle and making new triangles, which he compares with the others.
But is it credible that a thing so easy of proof as the equality of these angles required so much artifice to prove it, as though anything would be more ridiculous than to imagine that this equality depended on these foreign triangles, whereas, had the true order been observed, many other ways existed of proving this same equality, much shorter, easier, and more natural?
The 47th of the First Book, where it is proved that the square of the base which contains a right angle, is equal to the two squares of the sides, is one of the most admired propositions of Euclid: it is, however, sufficiently clear that the manner in which it is proved is not natural, since the equality of the squares does not depend on the equality of the triangles, which is taken as the mean in that demonstration, but on the proportion of the lines, which may be easily demonstrated without employing any other line than that let fall from the point of the right angle on the base.
All Euclid is full of these far-fetched demonstrations.
5th DEFECT.
Paying no attention to the true order of nature.
This is the greatest defect of the geometers.
They have fancied that there is scarcely any order for them to observe, except that the first propositions may be employed to demonstrate the succeeding ones. And thus, disregarding the true rule of method, which is, always to begin with things the most simple and general, in order to pass from them to those which are more complex and particular, they confuse everything, and treat pell-mell of lines and surfaces, and triangles and squares, proving by figures the properties of simple lines, and introducing a mass of other distortions which disfigure that beautiful science.
The elements of Euclid are quite full of this defect. After having treated of extension in the first four Books, he treats generally of the proportions of all kinds of magnitudes in the Fifth. He returns to extension in the Sixth, and treats of numbers in the Seventh, Eighth, and Ninth, and begins in the Tenth to speak again of extension. So much for the general disorder; but it is full of a mass of confusion in detail. He commences the First Book by the construction of an equilateral triangle, and, subsequently (twenty-two propositions after), he gives the general means for making any triangle of three given straight lines, provided that two are greater than a single one, which involves the particular construction of an equilateral triangle on a given line.
He proves nothing of perpendicular and parallel lines, except by triangles. He measures the dimension of surfaces with that of lines.
He proves—Book I., proposition 16—that, the side of a triangle being prolonged, the exterior angle is greater than either of the interior and opposite angles; and, six propositions further on, he proves that exterior angle is equal to the two opposite angles.
It would be necessary to transcribe the whole of Euclid, in order to give all the examples which might be found of this confusion.
6th DEFECT. Employing no divisions and partitions.
There is still another defect in the method of the geometers, that of not employing divisions and partitions. It is not that they do not mark all the species of the genera which they treat of, but that they do this simply by defining the term, and placing all the definitions one after another, without indicating that one genus has so many species, and can have no more, because the general idea of the genus can receive only so many differences, which would tend to throw considerable light upon the nature of genus and species.
For example, we find in the First Book of Euclid definitions of all the species of triangles. But who can doubt that it would be much clearer to speak of them as follows?—
A triangle may be divided in relation to its sides, or in relation to its angles.
It is, indeed, much better not to give the division of triangles before having explained and demonstrated the properties of triangle in general, from which we shall have learnt that at least two angles of a triangle must be acute, because the three together are only equal to two right angles.
The defect comes under that rule which enjoins that we do not treat of, or even define, the species before the genus is well known, especially when there are many things which may be said of the genus without speaking of the species.