geometrical constructions which, on inspection, suggested new theorems and invited scientific inquiry." 1

The Egyptians were also stimulated to a use of geometry for economic reasons. Herodotus2 tells us that Rameses II (cir. 1400 B.C.) divided the land of Egypt into equal squares so as to provide a more convenient means of taxation. But on account of the frequent overflows of the Nile, parts of these subdivisions were frequently swept away, and the king appointed surveyors to levy the proper tax in proportion to the part of the land remaining. There were certain benefits to be derived from the overflow of the Nile, and these again necessitated government supervision. Wilkinson says:3 'Besides the mere measurement of superficial areas, it was of paramount importance to agriculture. . . to distribute the benefits of the inundation in due proportion to each individual, that the lands which were low might not enjoy the exclusive advantages of the fertilizing water, by constantly draining it from those of a higher level. For this purpose, the necessity of ascertaining the various elevations of the country, and of constructing accurately levelled canals and dykes, obviously occurred to them. . . . These dykes were succeeded or accompanied by the invention of sluices, and all the mechanism appertaining to them; the regulation of the supply of water admitted into plains of various levels, the report of the exact quantity of land irrigated, the depth of the water admitted into plains of various levels, and the time it continued upon the surface, which determined the proportionate payment of the taxes, required much scientific skill." We thus see the development of a rudimentary surveying and with it the necessary practical geometry.

That the Egyptians considered their geometric learning to be worthy of preservation is shown from the contents of an old papyrus, which is now in the Rhind collection in the British Museum. This papyrus was written about 1700 B.C.

Gow, pp. 132-133.

2 Herodotus II, 109.

3 Wilkinson, The Ancient Egyptians, Vol. IV, pp. 7-8.

This was translated in 1877. See Eisenlohr, Ein Mathematisches Handbuch der alten Aegypter, pp. 125-150. There is a supposition that this is a copy of a much older work. For a copy of the original, see Facsimile of the Rhind Mathematical Papyrus in the British Museum. This contains a list of the principal works that describe the papyrus.

by the scribe Ahmes. Here are given rules for finding the areas of some of the plane figures, those for the areas of isosceles triangles and isosceles trapezoids being incorrect.1 Figures are given for these as well as for squares, rectangles, and circles. The diagrams are not lettered, but the lengths of various lines are indicated by means of the hieratic number symbols, which are copied on the figures just as is done to-day. As this is the oldest mathematical writing that has been deciphered, we have here the earliest known drawing of geometric figures employed for educative purposes. These incorrect rules in mensuration were used much later in some of the Egyptian temple inscriptions. The Temple of Horus at Edfu in Upper Egypt gives evidence of this fact. These inscriptions (cir. 100 B.C.) describe the lands "which formed the endowment of the priestly college attached to the temple." The incorrect formula for the isosceles trapezoid was also applied to any trapezoid. The a + b c + d formula for the general quadrilateral was

2

2

the four sides a, b, c, d are in general unequal.

2

where

Returning to the Ahmes papyrus, we find that there is given a formula for determining the area of a circle. Ahmes takes a circle whose diameter is 9 units and writes the area 64, using as a

formula (D

=

1

Area. This is equivalent to taking

D)

— D)2= Area.

9

π 3.1605, which shows a remarkably close approximation to the correct value. The papyrus also gives calculations for finding the contents of certain barns the shapes of which it does not accurately describe, and adds some examples on pyramids

1 Ahmes considers an isosceles triangle whose base is 4 units in length and whose leg is 10 units. The area is given as 20, showing that his formula is equivalent to multiplying the length of one of the equal legs by one-half the length of the base. He takes an isosceles trapezoid whose parallel bases are respectively 6 and 4 units each long. One of the equal legs is 4+ 6 20 units. The area is given equivalent to the form 2

X 20.

'There is in the British Museum a MS. written on leather, which, it is claimed, pertains to early Egyptian mathematics. It was sufficiently examined to convey this impression, but on account of the leather tending to crumble, it cannot be examined further.

'Gow, p. 131; Hankel, Zur Geschichte der Mathematik in Altertum und Mittelalter, pp. 86, 87, Hereafter referred to as Hankel,

which employ a rudimentary trigonometry in which the hypotenuse and base of right triangles are given to find their ratio (seqt). 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.'

92

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 Egypt1 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 (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.2

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."

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," 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

1 See below, p. 14.

2 Proclus, ed. Friedlein,

3 Ibid, p 352.

p. 299.

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

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