Commentary on Proposition II.10¶
Back to Proposition II.10.
The alternative proof of this proposition by means of the principles exhibited in II. 1—8 follows the lines of that which I have given for the preceding proposition.
It is at once obvious from the figure that the square on AD includes within it twice the square on AC together with once the square on CD. What is left over is the sum of the rectangles AH, KE. These, which are equivalent to BH, GK, make up the square on CD less the square on BD. Adding therefore the square BG to each side, we have the required result.
Another alternative proof of the theorem which includes both II. 9 and 10 is worth giving. The theorem states that the sum of the squares on the sum and difference of two given straight lines is equal to twice the sum of the squares on the lines.
Let AD, DB be the two given straight lines (of which AD is the greater), placed so as to be in one straight line. Make AC equal to DB and complete the figure as shown, each of the segments CG and DH being equal to AC or DB.
Now, AD, DB being the given straight lines, AB is their sum and CD is equal to their difference.
Also AD is equal to BC.
And AE is the square on AB, GK is equal to the square on CD, AK or FH is the square on AD, and BL the square on CB, while each of the small squares AG, BH, EK, FL is equal to the square on AC or DB.
We have to prove that twice the squares on AD, DB are equal to the squares on AB, CD.
Now twice the square on AD is the sum of the squares on AD, CB, which is equal to the sum of the squares BL, FH; and the figure shows these to be equal to twice the inner square GK and once the remainder of the large square AE excluding the two squares AG, KE, which latter squares are equal to twice the square on AC or DB.
Therefore twice the squares on AD, DB are equal to twice the inner square GK together with once the remainder of the large square AE, that is, to the sum of the squares AE, GK, which are the squares on AB, CD.
“Side” and “diagonal” numbers giving successive approximations to $\sqrt{2}$.¶
Zeuthen pointed out (Die Lehre von den Kegelschnitten im Altertum, 1886, pp. 27, 28) that II. 9, 10 have great interest in connexion with a problem of indeterminate analysis which received much attention from the ancient Greeks. If we take the straight line AB divided at C and D as in II. 9, and if we put CD $= x$, DB $= y$, the result obtained by Euclid, namely:
$$AD^2 + DB^2 = 2AC^2 + 2CD^2,$$
or
$$AD^2 - 2AC^2 = 2CD^2 - DB^2,$$
becomes the formula
$$(2x + y)^2 - 2(x + y)^2 = 2x^2 - y^2.$$
If therefore $x$, $y$ be numbers which satisfy one of the two equations
$$2x^2 - y^2 = \pm 1,$$
the formula gives us two higher numbers, $x + y$ and $2x + y$, which satisfy the other of the two equations.
Euclid’s propositions thus give a general proof of the very formula used for the formation of the succession of what were called “side” and “diagonal numbers.”
As is well known, Theon of Smyrna (pp. 43, 44, ed. Hiller) describes this system of numbers. The unit, being the beginning of all things, must be potentially both a side and a diameter. Consequently we begin with two units, the one being the first side and the other the first diameter, and ($a$) from the sum of them, ($b$) from the sum of twice the first unit and once the second, we form two new numbers
$$1 . 1 + 1 = 2, \quad 2 . 1 + 1 = 3.$$
Of these new numbers the first is a side- and the second a diagonal-number, or (as we may say)
$$a_2 = 2, \quad d_2 = 3.$$
In the same way as these numbers were formed from $a_1 = 1$, $d_1 = 1$, successive pairs of numbers are formed from $a_2$, $d_2$, and so on, according to the formula
$$a_{n+1} = a_n + d_n, \quad d_{n+1} = 2a_n + d_n.$$
Thus
$$\begin{aligned} a_3 &= 2 + 3 = 5, & d_3 &= 2 . 2 + 3 = 7, \\ a_4 &= 5 + 7 = 12, & d_4 &= 2 . 5 + 7 = 17, \end{aligned}$$
and so on.
Theon states, with reference to these numbers, the general proposition that
$$d_n^2 = 2a_n^2 \pm 1,$$
and he observes (1) that the signs alternate as successive $d$’s and $a$’s are taken, $d_1^2 - 2a_1^2$ being equal to $-1$, $d_2^2 - 2a_2^2$ equal to $+1$, $d_3^2 - 2a_3^2$ equal to $-1$, and so on, (2) that the sum of the squares of all the $d$’s will be double of the sum of the squares of all the $a$’s. [If the number of successive terms in each series is finite, it is of course necessary that the number should be even.] The proof, no doubt omitted because it was well known, may be put algebraically thus
$$\begin{aligned} d_n^2 - 2a_n^2 &= (2a_{n-1} + d_{n-1})^2 - 2(a_{n-1} + d_{n-1})^2 \\ &= 2a_{n-1}^2 - d_{n-1}^2 \\ &= -(d_{n-1}^2 - 2a_{n-1}^2) \\ &= +(d_{n-2}^2 - 2a_{n-2}^2), \text{ in like manner,} \end{aligned}$$
and so on, while $d_1^2 - 2a_1^2 = -1$. Thus the theorem is established.
Euclid’s propositions enable us to establish the theorem geometrically; and this fact might well be thought to confirm the conjecture that the investigation of the indeterminate equation $2x^2 - y^2 = \pm 1$ in the manner explained by Theon was no new thing but began at a period long before Euclid’s time. No one familiar with the truth of the proposition stated by Theon could have failed to observe that, as the corresponding side- and diagonal-numbers were successively formed, the value of $d_n^2 / a_n^2$ would approach more and more nearly to $2$, and consequently that the successive fractions $d_n / a_n$ would give nearer and nearer approximations to the value of $\sqrt{2}$, viz. $\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \dots$.
It is fairly clear that in the famous passage of Plato’s Republic (546 C) about the “geometrical number” some such system of approximations is hinted at. Plato there contrasts the “rational diameter of five” (ῥητὴ διάμετρος τῆς πεμπάδος) with the “irrational” (diameter). This was certainly taken from the Pythagorean theory of numbers (cf. the expression immediately preceding, 546 B, C πάντα προσήγορα καὶ ῥητὰ πρὸς ἄλληλα ἀπέφηναν, with the phrase πάντα γνωστὰ καὶ ποτάγορα ἀλλάλοις ἀπεργάζεται in the fragment of Philolaus). The reference of Plato is to the following consideration. If the square of side $5$ be taken, the diagonal is $\sqrt{2 . 25}$ or $\sqrt{50}$. This is the Pythagorean “irrational diameter” of $5$; and the “rational diameter” was the approximation $\sqrt{50 - 1}$, or $7$.
But the conjecture of Zeuthen, and the attribution of the whole theory of side- and diagonal-numbers to the Pythagoreans, have now been fully confirmed by the publication of Kroll’s edition of Procli Diadochi in Platonis rempublicam commentarii (Teubner), Vol. II., 1901. The passages (cc. 23 and 27, pp. 24, 25 and 27—29) which there saw the light for the first time describe the same system of forming side- and diagonal-*numbers and definitely attribute it, as well as the distinction between the “rational” and “irrational diameter,” to the Pythagoreans. :person:`Proclus <proclus>` further says (p. 27, 16—22) that the property of the *side- and diagonal-*numbers “is proved graphically (γραμμικῶς) in the second book of the Elements by *’him’ (ἀπ’ ἐκείνου). For, if a straight line be bisected and a straight line be added to it, the square on the whole line including the added straight line and the square on the latter alone are double of the square on the half of the original straight line and of the square on the straight line made up of the half and of the added straight line.” And this is simply Eucl. II. 10. Proclus then goes on to show specifically how this proposition was used to prove that, with the notation above used, the diameter corresponding to the side $a + d$ is $2a + d$. Let AB be a side and BC equal to it, while CD is the diameter corresponding to AB, i.e. a straight line such that the square on it is double of the square on AB. (I use the figure supplied by Hultsch on p. 397 of Kroll’s Vol. II.)
Then, by the theorem of Eucl. II. 10, the squares on AD, DC are double of the squares on AB, BD.
But the square on DC (i.e. BE) is double of the square on AB; therefore, by subtraction, the square on AD is double of the square on BD.
And the square on DF, the diagonal corresponding to the side BD, is double the square of BD.
Therefore the square on DF is equal to the square on AD, so that DF is equal to AD.
That is, while the side BD is, with our notation, $a + d$, the corresponding diagonal, being equal to AD, is $2a + d$.
In the above reference by Proclus to II. 10 ἀπ’ ἐκείνου “by him” must apparently mean ὑπ’ Εὐκλείδου, “by Euclid,” although Euclid’s name has not been mentioned in the chapter; the phrase would be equivalent to saying “in the second Book of the famous Elements.” But, when Proclus says “this is proved in the second Book of the Elements,” he does not imply that it had not been proved before; on the contrary, it is clear that the theorem had been proved by the Pythagoreans, and we have therefore here a confirmation of the inference from the part played by the gnomon and by I. 47 in Book II. that the whole of the substance of that Book was Pythagorean. For further detailed explanation of the passages of Proclus reference should be made to Hultsch’s note in Kroll’s Vol. II. pp. 393—400, and to the separate article, also by Hultsch, in the Bibliotheca Mathematica $\text{I}_3$, 1900, pp. 8—12.
Bergh has an ingenious suggestion (see Zeitschrift für Math. u. Physik
Hist.-litt. Abt. p. 135, and Cantor, Geschichte der Mathematik, $\text{I}_3$, p. 437) as to the way in which the formation of the successive side- and diagonal-*numbers may have been discovered, namely by observation from a very simple geometrical figure. Let :g:`ABC <0415.g1:< A B C >>` be an :eq:`isosceles <0415.g1:/ A B / = 0415.g1:/ A C />` triangle, right-angled at :g:`A <0415.g1:< B A C )>`, with sides $a_{n-1}$, $a_{n-1}$, $d_{n-1}$ respectively. If now the two sides :g:`AB <0415.g1:/ A B />`, :g:`AC <0415.g1:/ A C />` about the right angle be lengthened by adding $d_{n-1}$ to each, and the extremities :g:`D <0415.g1:D>`, :g:`E <0415.g1:E>` be joined, it is easily seen by means of the figure (in which :g:`BF <0415.g1:/ B F />`, :g:`CG <0415.g1:/ C G />` are perpendicular to :g:`DE <0415.g1:/ D E />`) that the new *diagonal $d_n$ is equal to $2a_{n-1} + d_{n-1}$, while the equal sides $a_n$ are, by construction, equal to $a_{n-1} + d_{n-1}$.
Important deductions from II. 9, 10.¶
Pappus (VII. pp. 856—8) uses II. 9, 10 for proving the well-known theorem that
The sum of the squares on two sides of a triangle is equal to twice the square on half the base together with twice the square on the straight line joining the middle point of the base to the opposite vertex.
Let ABC be the given triangle and D the middle point of the base BC. Join AD, and draw AE perpendicular to BC (produced if necessary).
Now, by II. 9, 10, the squares on BE, EC are equal to twice the squares on BD, DE.
Add to each twice the square on AE.
Then, remembering that the squares on BE, EA are equal to the square on BA, the squares on AE, EC are equal to the square on AC, and the squares on AE, ED are equal to the square on AD, we find that the squares on BA, AC are equal to twice the squares on AD, BD.
The proposition is generally proved by means of II. 12, 13, but not, I think, so conveniently as by the method of Pappus.
The inference was early made by Gregory of St. Vincent (1584–1667) and Viviani (1622–1703) that In any parallelogram the squares on the diagonals are together equal to the squares on the sides, or to twice the squares on adjacent sides.
It appears that Leonhard Euler (1707–83) was the first to discover the corresponding theorem with reference to any quadrilateral, namely that In any quadrilateral the sum of the squares on the sides is equal to the sum of the squares on the diagonals and four times the square on the line joining the middle
points of the diagonals. Euler seems however to have proved the property from the corresponding theorem for parallelograms just quoted (cf. Camerer’s Euclid, Vol. I. pp. 468, 469) and not from the property of the triangle, though the latter brings out the result more easily.