Addition and Subtraction

We choose freely the lines that are to be added or subtracted. We do not impose any definite quantity on them.

We now introduce rules, imposing a definite proportion on pairs of lines. These are not given in such a way that when they were combined they formed one of the 12 kinds of line already discussed.

But the pairs given in some other way, namely when it is required for one given line also to find its greater part so that the ratio of the smaller to the greater part will be equal to the ratio of the greater part to the sum of them both; or alternatively the ratio of the greater to the smaller will be equal to the ratio of the smaller to the remainder.

When the two are subtracted, the result will not always be a line belonging to some more remote degree of commensurability.

In these circumstances, we shall fall back on one of the kinds already discussed.

• We then compare the line that is found, which of itself is of the 8th degree, with lines of the 4th degree.

In this way, 2 lines of the 4th degree (as defined in Section 15) together formed an area, from which, when it is cast into the form of a square, there emerged as the side of the square a line called a Medial.

Thus, the 2 lines, the Whole and one part form:

• the other part by subtraction, or
  • In subtraction, the constituent lines were commensurable with one another only in square
• the whole by addition
  • In addition, in place of commensurability, we have identity of proportion between whole and parts.

The similarity of proportion was:

• between:
  • the smaller part and
  • the line to be formed [i.e. the difference]
• between:
  • the line to be formed and
  • the greater [part]

In the latter case:

• for subtraction, there is also a similarity of proportion:
  • between the 2 lines to be formed, and
  • between one of them and the proposed whole line [i.e. the sum], for subtraction
• for addition, the proportion is:
  • between one of the lines to be formed, and the proposed line, and the other line to be formed.

So, in the former case, given 2 lines, a Rectangle was given equal to the square of the line that was to be formed. Thus the area entered into the question before the line [i.e. the side of the square].

In the latter case, on the contrary, having made the 2 lines that were to be made, there then follows the equality between the Rectangle of the extreme lines and the square of the Mean one, by Euclid 6.17 and 11.11.4'

In the former case the straight lines that make the area had squares commensurable with the square of the proposed Line.

In the latter, from Euclid VL30 we know that we must take a square, commensurable with the square of the proposed line, nam ely! times it, and from the side of this square we must subtract half the proposed line, so that there remains the required part of the proposed line.

When this part is subtracted from the proposed line there is left the other required part, (or if it is added to the whole we obtain the third required line).

Parts that have so many terms should, it seems, be placed in the fourth degree.

At this point, we obtain a line of higher standing than the Medial itself, whichever line it is that the proportion is found iiU’: because the Medial hangs from the proposed Expressible line by a longer chain [of terms] made up of four links: whereas the parts of this [line] depend upon the ratio in which they stand directly to the proposed Expressible line.

From this it comes about that there can be many Medials all the same degree away from the Expressible line; indeed the larger part is the one and only part of an expressible line which is in this proportion, and in all cases of any line of lower standing than the Expressible there is one unique part in such proportion.

Because of this its construction is equivalent in a sense to the first degree.

Thus when the proposed Straight Line is required to be the whole, and we seek its two parts according to this proportion {tales), then Geometers call this division in Extreme and Mean ratio. Certainly this name means that whereas at other times ordinary division of the whole into two parts is not concerned with proportion, or if some line [is constructed] that bears to the whole the ratio of the smaller part to the greater, then there will result four terms, two extremes and two Means: in this case, on the contrary, there are only three terms, the whole and the smaller part being the two extremes; and the greater part being the unique mean term.

It is also, for the same reason, called proportional section.

Today both the section, and the proportion it defines, are given the title “Divine,” because of the marvelous nature of the section and its multiplicity of interesting properties: the foremost of which is that always when the greater part is added to the whole the compound line is again divided in the same way and the part which was the greater part now becomes the smaller part of the compound line; and the erstwhile whole line becomes the greater part of the compound line, by Euclid XIII.5.52