which employ a rudimentary trigonometry in which the hypotenuse and base of right triangles are given to find their ratio (seqt).1 This ratio determined the cosine of an angle, which, for all the pyramids, gives practically the same slant of the lateral faces. The term seqt was also used to denote the ratio which determines the tangent of an angle.

So much for the development of geometry by the Egyptians. They allowed the Greeks to come and learn, but just how this knowledge was communicated is hard to say. As the priests constituted the learned class, undoubtedly it came from them. Perhaps, also, the Greeks got some inspiration, as Gow relates, by observing the geometric constructions on the walls of the temples. Besides this, the architecture of the various temples would have given some instruction.

One could well ask why the Egyptians did not develop a logical geometry. Gow very properly states: "It will readily be supposed that the Egyptians, who had so early invented so many rules of practical geometry, could not fail in process of time to make many more discoveries of the same kind, and thus be led to geometrical science. But it appears that in Egypt, land-surveying, along with writing, medicine and other useful arts, was in the monopoly of the priestly caste; that the priests were the slaves of tradition, and that, in their obstinate conservatism, they were afraid to alter the rules or extend the knowledge of their craft. Of their medicine, Diodorus (I. 82) expressly relates that, even in his day, the Egyptian doctors used only the recipes contained in the ancient sacred books, lest they should be accused of manslaughter in case the patient died. Geometry seems to have been treated with similar timidity."

In brief, the Egyptians knew how to calculate the areas of some of the simple rectilineal figures, using some rules, however, that were erroneous. Also, they found the capacity of barns by methods not clearly defined. In some of their problems on pyramids the idea of ratio was involved. Finally, they employed some principles of symmetry in their mural decorations. On the whole, the Egyptians developed a practical geometry of areas.

1 Cantor, I, pp. 59-60; Gow, pp. 128-129. Cantor and Eisenlohr have worked out these interpretations.

2 Gow, p. 130.

As for the methods of instruction employed by the Egyptians, nothing is definitely known. With the exception of the manuscript of Ahmes, the undeciphered manuscript mentioned above, and some temple inscriptions, we have no record of Egyptian geometry from native sources.

THE GREEKS BEFORE EUCLID

The Development of the Subject-Matter of Elementary Geometry

The Greeks, who were the first to study geometry from a logical viewpoint, brought the subject into a coherent system during a period of 300 years. This development began when Thales in the capacity of a merchant visited Egypt' and there found the materials upon which to base his science, and culminated when Euclid (cir. 300 B.C.) wrote his "Elements. On his return to Asia Minor, Thales founded the Ionian School of mathematics and philosophy and was there visited by Pythagoras, who, after traveling in Egypt and perhaps Babylonia, founded his school at Croton in Southern Italy. When the Pythagorean school declined, after the death of its founder, the seat of learning was changed to Athens, which was then in the height of its power.

2

Geometry was studied arduously there by the Sophists, and, contemporaneous with them, Plato and his pupils contributed to the progress of the science. After the time of Aristotle we again see the study of geometry thrive on Egyptian soil at the newly founded city of Alexandria. In this Grecian city the logic of geometry was to be rounded out through the work of Euclid, whose text has influenced so largely the teaching of geometry even up to the present day. Let us now look into the additions to the subject-matter of geometry during this period of 300 years before Euclid, and also see what contributions were made from the standpoint of method.

Thales and his school have been credited with having added to geometry these five theorems:3 (1) A circle is bisected by its diameter; (2) the angles at the base of an isosceles triangle are

1 This is mentioned in the Eudemian Summary. See below p. 14. 2 Cantor, I, pp. 138-141; Gow, pp. 66, 148.

3 Cantor, I, pp. 124-136; Gow, pp. 140-145.

equal; (3) if two straight lines intersect the vertical angles are equal; (4) an angle inscribed in a semicircle is a right angle; (5) a triangle is determined if its base and base angles are known. That Thales did not prove all the propositions attributed to him is shown by a statement from the Eudemian Summary,' that Euclid first thought the third worthy of proof."

Although Thales was interested in the development of logical geometry, he was primarily an astronomer, and no doubt was impelled to further study of geometry by recognizing the relation between theory and practice. His practical turn of mind is referred to by Eudemus, who credited him with inventing a way of finding the distance of a ship at sea. The principle involved is associated with his fifth proposition mentioned above. Thales is also credited with finding the heights of pyramids by means of shadows.1

3

Enopides of Chios (cir. 450 B.C.), who seems to have been associated with the Ionic school, contributed to the development of geometry. According to Proclus,5 he solved the two problems: "From a point without a straight line of unlimited length to draw a straight line perpendicular to that line," and "At a given point in a given straight line to make an angle equal to a given angle." Concerning the first of these problems, Proclus says that Enopides first invented this problem, thinking it useful for astronomy. This is interesting, for it shows that the early Greeks did not entirely ignore practical geometry, and in particular we see the stimulating influence of science.

The work of Thales and the Ionian school was both practical and theoretical. While the Egyptians were concerned with areas in their practical work, Thales in his logical work developed theorems concerned with lines, which required a high degree of abstraction. To Thales, then, we can attribute the beginning of the geometry of lines and with it the deductive method of reasoning applied to geometry.

The next advance of any importance was made by Pythagoras, who seems to have been more directly influenced by the Egyp

4 Pliny, Natural History, trans. Bostock and Riley, xxxvi, 17

5 Proclus, ed. Friedlein, pp. 283, 333.

tians than was Thales. This is seen by the attention he gave to the geometry of areas and volumes, and to arithmetic. The school of Pythagoras was undoubtedly familiar with many of the propositions in the first two books of Euclid and with parts of the fifth and sixth books; that is, it was familiar with the ordinary theorems in plane geometry concerning equality of lines and of angles and with many of the theorems on equivalent and congruent areas. The geometry of the circle was not developed by the Pythagoreans, but was studied later at the Athenean school. It is said that the Pythagoreans knew that the angle sum of a triangle equals two right angles. Hence they held a conception of parallel lines if we are to suppose that any proof of the above theorem was accomplished. The theorem of the three squares (Euclid, I, 47), that in any right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, is known as the Pythagorean theorem. The Egyptians knew the truth of this theorem where the sides were in the ratio of 3:4:5, but Pythagoras was the first to see the truth of this relation in any right triangle.

Pythagoras is also credited with the discovery of the geometric irrational and of the three kinds of proportion, arithmetical geometric, and harmonic. The Pythagoreans were much interested in the study of the regular solids and are credited with their constructions. This being true, they were certainly familiar with the construction of the regular plane polygons of 3, 4, 5, sides. The construction of the regular polygon of five sides depends upon the division of a line in extreme and mean ratio. Allman' contends that Pythagoras was familiar with this, but Gow3 quotes from the Eudemian Summary, which attributes the discovery of this problem, known as the Golden Section, to the school of Plato. Gow's conclusion admits of less speculation and perhaps is nearer the truth.

We must note that in all that is known regarding the contributions of Thales and Pythagoras to the development of geometry there is more or less speculation as to the nature of the material, but in particular there is no definite statement re1 Gow, p. 153.

'Allman, Greek Geometry from Thales to Euclid, p. 40. Hereafter referred to as Allman

3 Op. cit., p. 153.

garding the methods of proof. We are, perhaps, safe in judging that a large part of the work of these early schools was in finding out geometric truths, and that the sequence of proved theorems was not then held in any hard and fast line.

We have no record that the school of Pythagoras was concerned with the practical, and we may conclude with Dr. Allman1 that the Pythagoreans were the first to sever geometry from the needs of practical life and to treat it as a liberal science.

Thus far we have the growth of geometry-first, the practical stage as with the Egyptians; second, the beginning of the logical stage in the school of Thales, where practical applications were employed and the foundations of deductive geometry laid; and thirdly, under the Pythagoreans, we have the subject treated as a liberal science. But we have no proof that the subjectmatter was yet organized into any fixed sequence.

When the Pythagorean school at Croton in Southern Italy was disbanded for political reasons, its influence had already grown and other schools had been founded on the shores of the Mediterranean. About this time, fresh from her glories of the Persian wars, Athens, exceedingly wealthy, attracted people of all nations. Among these were teachers who were willing to work for hire. Such were the Sophists. To them and the school of Plato, we are principally indebted for the great mass of subjectmatter which was finally organized into a text by Euclid of Alexandria. We recall that the Pythagoreans developed the geometry of areas but neglected the geometry of the circle. This study was taken up by the Athenian Greeks and many theorems were discovered in their futile attempts to solve the so-called Three Problems of Antiquity: the trisecting of any angle, the duplication of the cube, and the quadrature of the circle,2

Something of the nature of the contributions to the subjectmatter of geometry during this period can be seen from the titles of some of the works. Euclid is universally credited with being the first to write a complete text on geometry, but he was not the first to write on particular portions of it. Although the school of Thales is not generally credited with adding a great 1Op. cit., p. 47.

2 For a scientific treatment of these famous problems, see Klein, Famous Problems of Elementary Geometry, trans. Beman and Smith,