BOOK I. OF EUCLID'S " ELEMENTS."
§ 6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point given in the plane. This makes it possible (Prop. I) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and AB as radius, and then a circle with B as centre and BA as radius. The point where these circles intersect—that they intersect Euclid quietly assumes—is the vertex of the required triangle. Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance between any two points unless one of the points be the centre. This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane. This is done in two steps. It is shown in Prop. 2 how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point. And then the problem itself is solved in Prop. 3, by drawing first through the given point some straight line of the required length, and then about the same point as centre a circle having this length as radius. This circle will cut off from the given straight line a length equal to the required one. Nowadays, instead of going through this long process, we take a pair of compasses and set off the given length by its aid. This assumes that we may move a length about without changing it. But Euclid has not assumed it, and this proceeding would be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.
§ 7. We now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclid's whole system, there being only a very few propositions (like Props. 13, 14, 15, I.), except those in the 5th book and the first half of the 11th, which do not depend upon it. It is stated very accurately, though somewhat clumsily, as follows:
If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, namely, those to which the equal sides are opposite.
That is to say, the triangles are " identically " equal, and one may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other. It is then easily seen that also the remaining parts of one coincide with those of the other, and that they are therefore equal. This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so. The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.
If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop. 5), if two sides of a triangle are equal, then the angles opposite these sides are equal.
Euclid's proof is somewhat complicated, and a stumblingblock to many schoolboys. The proof becomes much simpler if we consider the isosceles triangle ABC (AB =AC) twice over, once as a triangle BAC, and once as a triangle CAB ; and now remember that AB, AC in the first are equal respectively to.AC, AB in the second, and the angles included by these sides are equal. Hence the triangles are equal, and the angles in the one are equal to those in the other, viz. those which are opposite equal sides, i.e. angle ABC in the first equals angle ACB in the second, as they are ,opposite the equal sides AC and AB in the two triangles.
There follows the converse theorem (Prop. 6). If two angles in, a triangle are equal, then the sides opposite them are equal,—i.e. the triangle is isosceles. The proof given consists in what is called a ' reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem. It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e. to a conclusion which is in contradiction to a proposition proved before—that therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of proof, but the method is most likely much older.
§ 8. It is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof.
These theorems allow now of the solution of a number of problems, viz.:
To bisect a given angle (Prop. 9).
To bisect a given finite straight line (Prop. Io).
To draw a straight line perpendicularly to a given straight line through a given point in it (Prop. II), and also through a given point not in it (Prop. 12).
The solutions all depend upon properties of isosceles triangles.
§ 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are ,together equal to two right angles. This theorem would have been unnecessary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.
Its converse (Prop. 14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the other. So also is
Prop. 15. If two straight lines cut one another, the vertical or opposite angles shall be equal.
§ 10. Euclid returns now to properties of triangles. Of great importance for the next steps (though afterwards superseded by a more complete theorem) is
Prop. 16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.
Prop. 17. Any two angles of a triangle are together less than two right angles, is an immediate consequence of it. By the aid of these two, the following fundamental properties of triangles are easily proved :
Prop. 18. The greater side of every triangle has the greater angle opposite to it;
Its converse, Prop. 19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it;
Prop. 20. Any two sides of a triangle are together greater than the third side;
And also Prop. 21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.
§ I r. Having solved two problems (Props. 22, 23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other. It is known (Prop. 4) that if the included angles are equal then the third sides are equal; and conversely (Prop. 8), if the third sides are equal, then the angles included by the first sides are equal. From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal. Euclid now completes this knowledge by proving, that " if the included angles are not equal, then the third side in that triangle is the greater which contains the greater angle "; and conversely, that " if the third sides are unequal, that triangle contains the greater angle which contains the greater side." These are Prop. 24 and Prop. 25.
§ 12. The next theorem (Prop. 26) says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, viz. in both triangles either the angles adjacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal.
This theorem belongs to a group with Prop. 4 and Prop. 8. Its first case might have been given immediately after Prop. 4, but the second case requires Prop. 16 for its proof.
§ 13. We come now to the investigation of parallel straight lines, i.e. of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The investigation which starts from Prop. 16, will become clearer if a few names be explained which are not all used by Euclid. If two straight lines be cut by a third, the latter is now generally called a " transversal " of the figure. It forms at the two points where it cuts the given lines four angles with each. Those of the angles which lie between the given lines are called interior angles, and of these, again, any two which lie on opposite sides of the transversal but one at each of the two points are called " alternate angles."
We may now state Prop. 16 thus:—If two straight lines which meet are cut by a transversal, their alternate angles are unequal. For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.
From this follows at once the theorem contained in Prop. 27. If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel. This proves the existence of parallel lines.
Prop. 28 states the same fact in different forms. If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make
EUCLIDEAN]
the interior angles on the same' side together equal to two right angles, the two straight lines shall be parallel to one another.
Hence we know that, " if two straight lines which are cut by a transversal meet, their alternate angles are not equal "; and hence that, " if alternate angles are equal, then the lines are parallel."
The question now arises, Are the propositions converse to these true or not ? That is to say, " If alternate angles are unequal, do the lines meet ?" And "if the lines are parallel, are alternate angles necessarily equal ?"
The answer to either of these two questions implies the answer to the other. But it has been found impossible to prove that the negation or the affirmation of either is true.
The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative. This gives his last axiom (12), which we quote in his own words.
Axiom 12.–If a straight line meet two straight lines, so as to make the two interior angles .on the' same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.
The answer to the second of the above questions follows from this, and gives the theorem Prop. 29:—If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.
§ 14. With this a new part of elementary geometry begins. The earlier propositions are. independent of this axiom, and would be true even if a wrong, assumption had been made in it. They all relate to figures in a plane. But a plane is only oneamong an infinite number of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead,of the plane, and obtain " spherical " in the place of " plane " geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all propositions up to the 28th will be true for these figures. This is the case in spherical geometry if we substitute " shortest line" or " great circle " for " straight line," " small circle " for " circle," and if, besides, we limit all figures to a part of the sphere which is less than a hemisphere, so that two points on it cannot be opposite ends of a diameter, and therefore determine always one and only one great circle.
For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and i8, 19 and 20 will hold good.
This remark will be sufficient to show the impossibility of proving Euclid's last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surface, where the other axioms do hold, whilst parallels do not even exist.
It follows that the axjqm in question states an inherent difference between the plane and other, surfaces, and that the plane is only fully characterized when this axiom is added to the other assump
i tonXs
§ 15. The introduction of, the new axiom and of parallel lines leads to a new class of propositions.
After proving (Prop. 30) that " two lines which are each parallel to a third are parallel to each other," we obtain the new properties of triangles contained in Prop. 32. Of these the second part is the most important, viz. the theorem, The three interior angles of every triangle are together equal to two right angles.
As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons; they are given in English editions as corollaries to Prop. 32.
These theorems do not hold for spherical figures. The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.
§ 16. The theory of parallels as such may be said to be finished with Props. 33 and J4, which state properties of the parallelogram, i.e. of a quadrilateral. formed by two pairs of parallels. They are
Prop. 33, The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel; and
Prop. 34. The opposite sides and angles of a parallelogram are equal to, one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.
§ 17. The rest of the first book relates to areas of figures. The theory is made to depend upon the theorems
Prop. 35. Parallelograms on the same base and between the same parallels are equal to one another; and
Prop. 36. Parallelograms on equal bases and between the same parallels are equal to one another.
As each parallelogram is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by " triangle," as is done in Props. 37 and 38.
It is to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the same straight line.
The theorems converse to the last form the contents of the next three propositions, viz.: Props, 40 and 4i.—Equal triangles, on679
the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.
That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.
§.18. To compare areas of other figures, Euclid shows first, in Prop. 42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is solved to draw a " rectangle " equal in area to a given triangle.
Next this parallelogram is transformed into another parallelogram, which has one of its sides equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in
Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.
Thus the problem (Prop. 44) is solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).
As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle. If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length. This is done in Prop. 43.
Herewith a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater. Euclid does not state thisconsequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.
Prop. 46 is: To describe a square on a given straight line.
§19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a rightangled triangle which is opposite the right angle the hypotenuse, we may state it as follows:
Theorem of Pythagoras (Prop. 47).—In every rightangled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.
And conversely
Prop. 48, If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.
On this theorem (Prop. 47) almost all geometrical measurement depends, which cannot be directly obtained.
End of Article: BOOK I 

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