Authors born between400 and 200 BCE
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Constructing an Equilateral Triangle
Rational and Irrational Quantities--Definitions
Euclid (c. 300 BCE) produced a comprehensive treatise on the mathematics of his time in a series of thirteen books known as the Elements. In totality, the books of Euclid constitute a superbly accurate theory of the physical space in which we live. The quality of this work is such it that survived as a teaching text for about 2,300 years, and dominated concepts of space and geometry until the Nineteenth Century. It is noteworthy for its rigorous approach of providing an initial set of definitions, common assumptions (axioms), and postulates, all of which appear to be obvious, and then deducing from them new findings that are not at all obvious. As such this became a model for further demonstrations in mathematics, such as Isaac Newton’s theory of gravitation and planetary motion, and for attempts to extend the procedure to other disciplines, such as the Ethics of Spinoza. Euclid thus made a major contribution to the process of reasoning whereby we seek to understand the world around us and ourselves.
Euclid also made due allowance for fallibility. In his discussion of parallel lines he made no attempt to hide his inability to prove a certain feature by putting it forward as a self-evident axiom. Instead he put it forward in the form of an elaborate supposition in a postulate. He was concerned with discovering the condition under which two lines would meet. He argued that one could test whether two lines would meet by crossing them with a third line. His assumption was that if the two interior angles formed on one side of this line added to less than two right angles, the two lines would eventually meet on that side (i.e. they would not be parallel). Many attempts were made to demonstrate that Euclid was mistaken and that this was a theorem that could be proved, rather than a supposition. All attempts failed until in early the Nineteenth Century when mathematicians such as Gauss and Lobachevsky found that different suppositions led to different geometries, notably the non-Euclidian geometries dealing with astronomically large spaces considered in the general theory of relativity.
Euclid may have been active around 300 BCE, because there is a report that he lived at the time of the first Ptolemy, and because a reference by Archimedes to Euclid indicates he lived before Archimedes (287-212 BCE). Euclid is likely to have gained his mathematical training in Athens, from pupils of Plato. Many of the older mathematicians on whose work Euclid’s Elements depends lived and taught there.
Euclid also wrote about astronomy, music and optics, but is most famous for his school of mathematics at Alexandria, where he taught. In his Elements he included theorems of earlier mathematicians, notably Eudoxus and Theaetetus, and also his own rigorous demonstrations of theorems that were only loosely proved by his predecessors. He was scrupulous in acknowledging the work of others. His Books I, II, IV cover lines, areas and plane figures; Book III deals with circles. Book V expands on the work of Eudoxus on proportion, providing a basis for discussion of similar figures in Book VI. Books VII, VIII, and IX deal with arithmetic and the theory of numbers. Book X deals with irrational numbers, which cannot be expressed as a simple ratio between two integers. These had perplexed the Pythagoreans when they were often found necessary to represent the largest side of a right-angled triangle. Eudoxus made a major discovery in arithmetic when he showed how they can be handled, and Euclid elaborated on this work. Book XI covers elementary solid geometry; Book XII formally proves the theorem of Hippocrates (not the practitioner of healing) for the area of a circle—pi times the radius squared. Book XIII provides and proves the constructions for the five regular solids of Pythagoras.
The extracts below illustrate some of Euclid’s definitions, postulates, and common notions, and how they are applied to inscribing a triangle in a circle and to proving the Pythagorean theorem that the square on the longest side of a right angled triangle is equal to the sum of the squares on the other two sides. The practical results of this theorem were known to the Egyptians. It meant that ropes with lengths of three, four and five units (a Pythagorean triad) could be formed into a right-angled triangle for the purpose of defining field boundaries. The fact that three-squared plus four-squared equals five-squared was known to the Egyptians as far back as 2000 BCE.
Similar knowledge appeared early in India. In showing how to construct altars of certain shapes, the Apastamha Sulvasutras contain rules for constructing rectangles involving the use of the diagonal, and enunciates a relationship for the rectangle that is similar to the Pythagorean theorem. The date these rules were written down is put at the fourth or fifth century BCE, but the knowledge may come from a much earlier period. The construction of a right angle from ropes of appropriate lengths may date back to the 8th century BCE in India. There is also a Chinese work, attributed to Chou Kung (died about 1100 BCE), that provides a rule for finding the hypotenuse of a right-triangle from the lengths of the other two sides, and states that the diagonal of a rectangle with sides three units and four units is five units. However, it was left to the Greeks to prove that in any right angled triangle the square on the longest side was equal to the sum of the squares on the other two sides.
It is noteworthy that Euclid’s first geometric definition, that a point has no part, i.e. has no dimensions, has raised a problem in particle physics, where the electron and other fundamental particles appear to have the dimensions of just such a point. The problem is that if they do, any property they have—such as electric charge—has in infinite rate of change at that point. This leads to wildly unrealistic infinities when theories of their behavior are investigated in mathematical terms. A mathematical technique called renormalization is used to get rid of these infinities, but its arbitrary nature causes some uneasiness, and one can imagine Euclid shaking his head over it. Possibly there is a limit to smallness and that a point, while having no part as Euclid assumed, may have a finite but unimaginably small size. This is another example of the richness of the ideas that Euclid pondered over centuries ago.
1. A point is that which has no part.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The extremities of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.
8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
9. And when the lines containing the angle are straight, the angle is called rectilineal.
10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
11. An obtuse angle is an angle greater than a right angle.
12. An acute angle is an angle less than a right angle.
13. A boundary is that which is an extremity of anything.
14. A figure is that which is contained by any boundary or boundaries.
15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
16. And the point is called the center of the circle.
17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.
19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
2l. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Let the following be postulated:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
Proposition 1. On a given finite straight line to construct an equilateral triangle.
Let AB be the given finite straight line.
Thus it is required to construct an equilateral triangle on AB.
With center A and distance AB let the circle BCD be described. [Post. 3]
With center B and distance BA let the circle ACE be described. [Post. 3]
And from the point C, in which the circles cut one another,
to the points A, B let the straight lines CA, CB be joined. [Post 1]
Now, since the point A is the center of the circle CDB,
AC is equal to AB. [Def. 15]
Again, since the point B is the center of the circle CAE,
BC is equal to BA. [Def. 15]
But CA was also proved equal to AB;
therefore each of the straight lines CA, CB is equal to AB.
And things which are equal to the same thing are also
equal to one another; [C. N. 1]
therefore CA is also equal to CB.
Therefore the three straight lines CA, AB, BC are equal to one another.
[Therefore the triangle ABC is equilateral.]
Book I. 1
Proposition 47 In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Let ABC be a right-angled triangle having the angle BAC right;
I say that the square on BC is equal to the squares on BA, AC.
For let there be described on BC the square BDEC,
and on BA, AC the squares GB, HC [I. 46]
Through A let AL be drawn parallel to either BD or CE,
and let AD, FC be joined.
Then, since each of the angles BAC, BAG is right,
it follows that with a straight line BA, and at the point A on it,
the two straight lines AC, AG not lying on the same side
make the adjacent angles equal to two right angles.
Therefore CA is in a straight line with AG. [I. 14]
For the same reason BA is also in a straight line with AH.
And, since the angle DBC is equal to the angle FBA, for each is right,
let the angle ABC be added to each.
Therefore the whole angle DBA is equal to the whole angle FBC. [C.N. 2]
And, since DB is equal to BC, and FB to BA,
the two sides AB, BD are equal to the two sides FB, BC respectively,
and the angle ABD is equal to the angle FBC.
Therefore the base AD is equal to the base FC,
and the triangle ABD is equal to the triangle FBC. [ I. 4]
Now the parallelogram BL is double of the triangle ABD,
for they have the same base BD and
are in the same parallels BD, AL. [I. 41]
And the square GB is double of the triangle FBC,
for they again have the same base FB and
are in the same parallels FB, GC. [I. 41]
[But the doubles of equals are equal to one another.]
Therefore the parallelogram BL is also equal to the square GB.
Similarly, if AE, BK be joined, the parallelogram CL
can also be proved equal to the square HC.
Therefore the whole square BDEC is equal to the two squares GB, HC. [C. N. 2]
And the square BDEC is described on BC,
and the squares GB, HC on BA, AC.
Therefore the square on the side BC is equal to the squares on the sides BA, AC.
Therefore in right-angled triangles the square on the side subtending the right angle
is equal to the squares on the sides containing the right angle. Q.E.D.
Book I. 47
1. Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable
cannot have any common measure.
2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.
3. With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in
length and in square or in square only, rational, but those which are incommensurable with it irrational.
4. And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.
The Thirteen Books of Euclid’s Elements, translated by Thomas L. Heath. Cambridge University Press, Cambridge, England, 1908
Tufts University Perseus Project
Clark University, Department of Mathematical and Computer Sciences