Institutio Oratoria

As regards geometry, [*](Geometry here includes all mathematics.) it is granted that portions of this science are of value for the instruction of children: for admittedly it exercises their minds, sharpens their wits and generates quickness of perception. But it is considered that the value of geometry resides in the process of learning, and not as with other sciences in the knowledge thus acquired. Such is the general opinion.

But it is not without good reason that some of the greatest men have devoted special attention to this science. Geometry has two divisions; one is concerned with numbers, the other with figures. Now knowledge of the former is a necessity not merely to the orator, but to any one who has had even an elementary education. Such knowledge is frequently required in actual cases, in which a speaker is regarded as deficient in education, I will not say if he hesitates in making a calculation, but even if he contradicts the calculation which he states in words by making an uncertain or inappropriate gesture with his fingers. [*]( There was a separate symbol for each number, depending on the hand used and the position of the fingers. See Class. Review, 1911, p. 72 ) Again linear geometry is frequently required in cases, as in lawsuits about boundaries and measurements.

But geometry and oratory are related in a yet more important way than this.

In the first place logical development is one of the necessities of geometry. And is it not equally a necessity for oratory? Geometry arrives at its conclusions from definite premises, and by arguing from what is certain proves what was previously uncertain. Is not this just what we do in speaking? Again are not the problems of geometry almost entirely solved by the

For, if necessary, he will use the syllogism, and he will certainly make use of the enthymeme which is a rhetorical form of syllogism. [*]( See v. xiv. I for an example from the Pro Ligario. The cause was then doubtful, as there were arguments on both sides. Now, however, we must regard that cause as the better, to which the gods have given their approval. ) Further the most absolute form of proof is that which is generally known as linear demonstration. And what is the aim of oratory if not proof?

Again oratory sometimes detects falsehoods closely resembling the truth by the use of geometrical methods. An example of this may be found in connexion with numbers in the so-called pseudographs, a favourite amusement in our boyhood. [*](It is not known to what Quintilian refers.) But there are more important points to be considered. Who is there who would not accept the following proposition?

When the lines bounding two figures are equal in length, the areas contained within those lines are equal.

and historians have been taken to task by geometricians for believing the time taken to circumnavigate an island to be a sufficient indication of its size. For the space enclosed is in proportion to the perfection of the figure.

Consequently if the bounding line to which we have referred form a circle, the most perfect of all plane figures, it will contain a greater space than if the same length of line took the form of a square, while a square contains a greater space than a triangle having the same total perimeter, and an equilateral triangle than a scalene triangle.

But there are other points which perhaps present greater

But a square of 180 feet gives the same perimeter, yet contains a much larger area within its four sides. If the calculation prove irksome to any of my readers, he can learn the same truth by employing smaller numbers. Take a ten foot square: its perimeter is forty feet and it contains 100 square feet. But if the dimensions be fifteen feet by five, while the perimeter is the same, the area enclosed is less by a quarter.

On the other hand if we draw a parallelogram measuring nineteen feet by one, the number of square feet enclosed will be no greater than the number of linear feet making the actual length of the parallelogram, though the perimeter will be exactly as that of the figure which encloses an area of 100 square feet. Consequently the area enclosed by four lines will decrease in proportion as we depart from the form of a square.

It further follows that it is perfectly possible for the space enclosed to be less, though the perimeter be greater. This applies to plane figures only: for even one who is no mathematician can see that, when we have to consider hills or valleys, the extent of ground enclosed is greater than the sky over it.

But geometry soars still higher to the consideration of the system of the universe: for by its calculations it demonstrates the fixed and ordained courses of the stars, and thereby we acquire the knowledge that all things are ruled by order and destiny, a consideration which may at times be of value to an orator.

When

If Nicias had known this when he commanded in Sicily, he would not have shared the terror of his men nor lost the finest army that Athens ever placed in the field. Dion for instance when he came to Syracuse to overthrow the tyranny of Dionysius, was not frightened away by the occurrence of a similar phenomenon. However we are not concerned with the uses of geometry in war and need not dwell upon the fact that Archimedes singlehanded succeeded in appreciably prolonging the resistance of Syracuse when it was besieged.

It will suffice for our purpose that there are a number of problems which it is difficult to solve in any other way, which are as a rule solved by these linear demonstrations, such as the method of division, section to infinity, [*]( Quintilian is perhaps referring to the measurement of the area of an irregular figure by dividing it into a number of small equal and regular figures the size of which was calculable. ) and the ratio of increase in velocity. From this we may conclude that, if as we shall show in the next book an orator has to speak on every kind of subject, he can under no circumstances dispense with a knowledge of geometry.