Return to QRV-321 Classical Geometry I
In this lesson, we will study Euclid, Book I, Proposition 1.
I. ENUNCIATION
The proposition we are now to contemplate is the first of the Elements. Euclid states it in these words:
On a given finite straight line to construct an equilateral triangle.
Let us first understand the words themselves, before we ask why we should care to know this.
A straight line, in Euclid’s usage, is not what we today call a “line” stretching endlessly. It is what we would call a segment: a length with two ends. He calls it finite to make that plain — it is bounded, it has two extremities, and it does not go on without limit. So “a given finite straight line” means: some particular straight length, already there before us, with two ends, which we did not make and do not choose.
Equilateral means “equal-sided.” An equilateral triangle, by Definition 20, is a three-sided figure “which has its three sides equal.”
To construct something on a given straight line means to produce it so that the given line is one of its sides. We are not asked to make an equilateral triangle out of nothing, nor one of whatever size we please. We are asked to make one upon this line, the line handed to us.
The purpose of this study
Before the purpose of this proposition, consider the purpose of Geometry.
Geometry is a speculative science. That is, its end is not to make anything or to accomplish any work, but simply to know. Its objects — points, lines, angles, figures — are not found in some separate world apart from the things we see and touch. They are found in the very things we see and touch, and the mind reaches them by abstraction: the geometer looks at the same bronze ring the smith looks at, but he considers only its roundness, its magnitude, its figure, setting aside the bronze. Knowledge begins in the senses. You will learn nothing in this course by turning away from the visible world; you will learn it by considering the visible world in a certain way.
But what you then hold is no longer a fact about any visible thing. When you have demonstrated that a triangle so constructed has three equal sides, you hold something that is true always and everywhere, and that could not have been otherwise. No ruler can repeal it; no new observation can overturn it; it was true before you were born and will be true when this world has passed. Plato said in the seventh book of the Republic that the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient, and that geometry will draw the soul towards truth and create the spirit of philosophy. That is the end of this study. The order and measure the geometer finds in things is the order God placed in them, and the eternal necessity of a demonstrated conclusion is a real participation in the truth of the Eternal.
And here is how that spirit of philosophy is actually created in you. Aristotle teaches that we know a thing, in the strict sense, when we know its cause, and know that the thing cannot be otherwise. In most of what you are taught in life, you are told a conclusion and asked to accept it. In Euclid you are shown the causes, and you arrive at a conclusion you cannot doubt, and you know why you cannot doubt it. Having once had that experience, you possess a standard. Ever afterward, when anything is asserted to you in any subject whatever, you can hold it against that standard and ask: is this demonstrated, or merely asserted? Do I see the cause, or have I only been told the fact? Most men never acquire this standard and therefore cannot tell knowledge from opinion. Euclid gives it.
Now, what does this proposition offer? Three things.
First, it is the place where you cross from principles to knowledge. Up to this point you have been given definitions, postulates, and common notions — things granted, not proved. Here, for the first time, something is obtained from them. You will see with your own reason how a conclusion is drawn out of principles, and you will see that the whole strength of the conclusion comes from the principles and from nothing else.
Second, it gives us the equilateral triangle — not the drawing of one, but the certain knowledge that upon any straight line whatever, however long or short, wherever it lies, such a triangle can be made. This is knowledge of a possibility that holds universally.
Third, and this matters more than it may seem: the proposition teaches you that in geometry we may not simply assume that a thing exists because we can name it and define it. Definition 20 tells us what an equilateral triangle is. It does not tell us that there is one. Euclid will not use a figure in a demonstration until he has shown that it can be produced from what has been granted. That discipline — never to help yourself to what you have not established — is the whole discipline of demonstrative science, and this proposition is where it begins.
So the student who finishes this lesson should not think the point was the triangle. The point was the demonstration.
Where we now stand
Nothing has yet been proved. Everything we have is given to us at the head of Book I, and it is worth gathering it briefly, because in this proposition we shall be permitted to use nothing else.
We are given twenty-three definitions. The first seven fix the elements of magnitude: a point has no part; a line is breadthless length; the extremities of a line are points; a straight line lies evenly with the points on itself; a surface has length and breadth only; the extremities of a surface are lines; a plane surface lies evenly with the straight lines on itself. Definitions 8 through 12 concern angles: an angle is the inclination of two lines that meet and do not lie in one straight line; it is called rectilineal when both lines are straight; the right angle, the perpendicular, the obtuse and the acute are defined by comparison to it. Definitions 13 and 14 give us boundary and figure. Definitions 15 through 18 give the circle, its centre, its diameter, its semicircle. Definitions 19 through 22 give the rectilineal figures — the triangles by their sides and by their angles, and the quadrilaterals. Definition 23 gives parallel straight lines.
We are given five postulates — things we ask leave to do or to accept, which cannot be proved and are granted at the outset: to draw a straight line from any point to any point; to produce a finite straight line continuously; to describe a circle with any centre and distance; that all right angles are equal to one another; and the fifth, concerning lines that must meet when a line falling across them makes the interior angles on one side less than two right angles.
We are given five common notions — truths not proper to geometry alone but common to all reasoning about magnitudes: things equal to the same thing are equal to one another; if equals be added to equals the wholes are equal; if equals be subtracted from equals the remainders are equal; things which coincide with one another are equal to one another; the whole is greater than the part.
That is our whole stock. From it, and from nothing else, this proposition must be got.
Problem or theorem
Every proposition in the Elements is either a problem or a theorem.
A problem requires something to be constructed or done, and it closes with the words which was to be done.
A theorem asserts that something is true of what is already given, and it closes with the words which was to be demonstrated.
This proposition is a problem. Notice the form of the enunciation: not “an equilateral triangle has such a property,” but “to construct an equilateral triangle.” Something is to be brought into being out of what we were given. Therefore the lesson will end with which was to be done, and the certitude you acquire will be certitude that a thing can be done, and that when it is done it necessarily has the property claimed.
II. EXPOSITION
Euclid now takes the general statement and applies it to a particular case, naming the parts with letters. He says:
Let AB be the given finite straight line.
Consider first why we use letters at all. Geometry must speak with precision, and ordinary speech cannot say “that line, the one over there” without ambiguity. So we mark the ends of a straight line with letters, and thereafter the line has a name. A straight line is named by its two extremities, which are points: the line whose ends are the points A and B is called the line AB, or equally the line BA, for it is the same line taken from the other end. An angle is named by three letters, the middle one being the point of the angle. A circle is named by letters marking points on the line that bounds it. A triangle is named by its three corners.
So: let there be some straight line, and let one of its ends be called A and the other B. That line is now the line AB, and we can speak of it without confusion.
What is known. This is where our investigation begins, and I ask you to be very exact about it, for a great part of learning Euclid consists in refusing to assume what has not been given. What is given is:
one finite straight line, with two extremities, named A and B.
That is all. And now let me warn you of what is not given:
We are not told how long AB is. It may be any length. Nothing in what follows may depend on its length.
We are not told where AB lies, nor in what direction it runs. Nothing may depend on that.
We are not given any other point, line, circle, or figure anywhere. If we want anything else, we must produce it by a postulate or by a proposition already demonstrated — and no proposition has yet been demonstrated, so here it must be by a postulate.
We are not given a ruler marked with lengths, nor any means of measuring. Euclid never measures. He compares.
Everything else in this lesson must come from the twenty-three definitions, the five postulates, and the five common notions. If at any step you find yourself granting something for another reason — because it seems reasonable, or because a drawing appears to show it — stop, because at that moment the certitude is lost.
III. SPECIFICATION
Euclid states the goal for this particular line:
Thus it is required to construct an equilateral triangle on the straight line AB.
What is unknown, that is, what is sought, is this: a third point, not on the line AB, such that when it is joined by straight lines to A and to B, the three-sided figure so contained has its three sides equal to one another.
Let us see the shape of the whole inquiry, so that you know what must be crossed.
We hold one line, AB.
We seek a figure of three sides, all equal.
Two of the three sides are easy to see in advance: AB itself will be one side, and the other two will be the straight lines drawn from the new point to A and to B. Postulate 1 will let us draw those two lines as soon as we have the point. So the whole difficulty lies in one thing only: finding a point whose distance from A is the same as the length AB, and whose distance from B is also the same as the length AB.
That is the distance to be crossed. And notice the real difficulty in it. We have no means of measuring the length AB and carrying that measure elsewhere. We are not permitted to say “take a length equal to AB and set it here,” because nothing granted to us permits it. (Indeed, the very next proposition, I.2, exists in order to establish that a length may be transferred, and it depends on the present proposition, so we certainly cannot use it now.)
So the whole art of the construction is to find a way of producing a point at a determined distance from a given point, using only what we have. Postulate 3 and Definition 15 together will do it, and you should watch how.
This proposition carries no condition of possibility. Some later problems can only be done if the things given satisfy some requirement — as in I.22, where three straight lines will only make a triangle if any two of them together are greater than the third. Here there is no such restriction. Any finite straight line whatever will serve. This is worth noticing, because it means the conclusion will be entirely universal.
IV. CONSTRUCTION
Euclid’s construction has three steps. I will give each one, and for each one state three things: what is done, what permits it to be done, and what we now have that we did not have before.
Step 1
With centre A and distance AB let the circle BCD be described. [Post. 3]
What is done. We describe a circle. Its centre is the point A — that is, by Definition 16, A is the point from which all the straight lines falling upon the circle’s boundary are equal. Its distance is AB — Euclid’s word for what we now call the radius, the length from the centre out to the boundary. So we describe the circle which has A for its centre and which passes through B.
What permits it. Postulate 3: “To describe a circle with any centre and distance.” We have a point, A, to serve as centre, and we have a length, AB, to serve as distance. Both were given. So the postulate grants this circle without further argument. Notice that we did not have to make the length AB; we already had it, since the given line is AB itself. This is exactly why the construction is possible at all.
The naming. Euclid calls it “the circle BCD.” Three letters, none of which is A, the centre. This is how a circle is named: by points lying on the line that bounds it, called the circumference. B is one such point, since the circle passes through B. C and D are further points on it, which Euclid marks now so that he can speak of them later. Do not be troubled that the circle is here called BCD and a few lines later CDB; a circle is named by naming points on it, and the order is indifferent. Restated plainly: let a circle be described about the point A as centre, passing through the point B.
What we now have. A circle whose centre is A and which passes through B. By Definition 15, every straight line drawn from A to any point of that circle’s boundary is equal to every other such line — and therefore equal to AB, which is one of them.
Step 2
Again, with centre B and distance BA let the circle ACE be described. [Post. 3]
What is done. The same thing, with the two points exchanged. We describe the circle whose centre is B and whose distance is BA — the same length as AB, since it is the same line named from the other end. This circle therefore passes through A.
What permits it. Postulate 3 again, word for word. We have a point, B, and a length, BA. Nothing more is needed.
The naming. “The circle ACE”: A is on it, because the circle passes through A; C and E are further points marked on its circumference.
What we now have. A second circle, whose centre is B, and every straight line from B to its boundary is equal to BA.
Step 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]
What is done. Euclid takes C to be a point where the two circles cut one another — that is, a point lying on the boundary of the first circle and also on the boundary of the second. From that point he joins straight lines to A and to B. To “join” two points means simply to draw the straight line from one to the other; the line CA is the straight line from C to A, and CB the straight line from C to B.
What permits the joining. Postulate 1: “To draw a straight line from any point to any point.” Once we have the point C, and we have the points A and B, this postulate grants us both straight lines at once, without any further reason.
An honest word about the point C. Euclid says “the point C, in which the circles cut one another,” and he does not prove that they cut one another. He assumes it. I tell you plainly that this is an assumption not licensed by any of his definitions, postulates, or common notions, and it is the one weak joint in this first proposition. Later geometers, examining the Elements with care, supplied what is called a principle of continuity, to the effect that a continuous line passing from the inside of a circle to the outside must meet the circle. Euclid took such matters as belonging to the nature of continuous magnitude and did not set them down.
I draw your attention to it for a reason that serves the end of this course. You must not grant this point because a drawing appears to show two circles crossing. What a pencil draws is not the geometrical thing at all: a geometrical line has no breadth and cannot be seen, while the drawn line has breadth and is seen. If your certitude about C rests on the appearance of a drawing, then it is not certitude. Here, then, is precisely where you should notice the difference between what is demonstrated and what is assumed — and having noticed it, you have already begun to acquire the standard this course exists to give you. For the remainder of the lesson we proceed on the assumption Euclid makes, and I shall say no more about it.
What we now have. A point C lying on both circles, and two straight lines CA and CB. Together with the given line AB, these three straight lines contain a three-sided figure, which by Definition 19 is a trilateral figure, that is, a triangle: the triangle ABC.
The whole construction restated plainly
Take the given line, with its ends A and B. Describe a circle about A passing through B. Describe a circle about B passing through A. These two circles cut one another; call one of the points where they cut C. Draw the straight line from C to A, and the straight line from C to B.
Now that the whole of it has been explained, you may take up your own compasses and draw it. But understand what you are then doing: you are exercising knowledge you already hold, not acquiring it. The reasoning has been complete in words, and your drawing adds nothing to it.
V. DEMONSTRATION
We have a triangle. It remains to prove that its three sides are equal — that it is equilateral. Every step rests on Definition 15 or on Common Notion 1, and on nothing else.
Let us first have Definition 15 exactly before us, since the whole proof turns on it:
“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; and the point is called the centre of the circle.”
Read that slowly. It says that a circle is bounded by a single line, and that there is a point within it — the centre — from which every straight line drawn to that bounding line is equal to every other. This is not a property of circles discovered by some later argument. It is what the word circle means. To grant that a figure is a circle with centre A is already to grant that all straight lines from A to its boundary are equal. Whoever refuses this is not disagreeing with a demonstration; he is refusing to use the word.
First step: CA is equal to AB
Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15]
Set this out as an argument, so you may see the two premises and the conclusion.
First premise. By Definition 15, all straight lines drawn from the centre of a circle to the line bounding it are equal to one another.
Second premise. A is the centre of this circle — not by observation, but by our construction, for in Step 1 we described the circle with A as centre, and Postulate 3 granted it with A as centre. And both C and B lie on the line bounding that circle: B, because the circle was described through B; C, because C was taken as a point at which this circle is cut by the other, and a point where a circle is cut is a point on the circle.
Conclusion. Therefore AC and AB are two straight lines drawn from the centre of one and the same circle to its bounding line. Therefore, by the definition, AC is equal to AB.
Observe the necessity. Could AC have failed to equal AB? Only if the figure we described were not a circle with centre A. But it is such a circle by the very postulate under which we described it. So once the construction is granted, the equality cannot be refused. It is not likely, or usual, or true of most cases: it could not be otherwise.
Second step: BC is equal to BA
Again, since the point B is the centre of the circle CAE, BC is equal to BA. [Def. 15]
The argument has exactly the same form, and I set it out again so that you may hear it as an argument and not merely as a repetition.
First premise. By Definition 15, all straight lines from the centre of a circle to its bounding line are equal to one another.
Second premise. B is the centre of the second circle, by our construction in Step 2. And A lies on its boundary, since the circle was described through A; and C lies on its boundary, since C was taken as a point in which this circle is cut by the other.
Conclusion. Therefore BC and BA are both drawn from the centre B to the boundary of one and the same circle. Therefore BC is equal to BA.
Notice how the second circle earns its place. Had we described only one circle, we should have had a point C at the right distance from A, but nothing whatever would fix its distance from B. The second circle is not a duplicate; it is what supplies the second of the two conditions we said in the Specification had to be met.
Third step: gathering what has been proved
But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB.
Here Euclid simply collects the two results. From the first step, CA is equal to AB. From the second step, CB is equal to BA — and BA is the very same straight line as AB, named from the other end, so CB is equal to AB.
I supply that last remark because a beginner may not supply it himself: BA and AB are two names for one line, not two lines. Euclid passes over it in silence; we should not.
So now we have: CA is equal to AB, and CB is equal to AB. Two lines, each of them equal to one and the same third line.
Fourth step: CA is equal to CB
And things which are equal to the same thing are also equal to one another; therefore CA is also equal to CB. [C.N. 1]
First premise. Common Notion 1: “Things which are equal to the same thing are also equal to one another.” This is granted at the outset. It is not proper to geometry alone but holds of all magnitudes whatever, and it cannot be demonstrated from anything simpler, because there is nothing simpler.
Second premise. CA is equal to AB, and CB is equal to AB. That is, CA and CB are both equal to the same thing, namely AB.
Conclusion. Therefore CA is equal to CB.
Consider what has just happened, because it is the pattern of all demonstration. We never compared CA with CB directly. We had no way to do so. Instead we found a third magnitude, AB, to which each of them could be shown equal, and through that third magnitude the equality of the two was reached. In the logic of Aristotle this third thing is called the middle term: the term through which two others are joined. Demonstration is not the assertion of conclusions; it is the finding of middle terms. Here the middle term is the line AB — the very thing that was given us at the start.
Fifth step: the conclusion about the triangle
Therefore the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB.
First premise. CA is equal to AB (first step); CB is equal to AB (second step); CA is equal to CB (fourth step). So the three straight lines CA, AB, BC are equal, each to each.
Second premise. Definition 20: an equilateral triangle “is that which has its three sides equal.” And by Definition 19, a figure contained by three straight lines is a trilateral figure, a triangle. The figure ABC is contained by the three straight lines CA, AB, BC.
Conclusion. Therefore ABC is a triangle whose three sides are equal, that is, an equilateral triangle. And one of its sides is AB, the line that was given us; therefore it has been constructed on the given finite straight line, as was required.
Nothing remains. What was sought has been produced, and its property has been proved from the definitions of a circle and of an equilateral triangle, from the two postulates that permitted the circles and the joining lines, and from the first common notion.
VI. CONCLUSION
Return now to the universal statement with which we began: on a given finite straight line to construct an equilateral triangle.
It now stands accomplished — and not for the line AB alone. Look back over every step and ask what was used. The length of AB was never used. Its position was never used. Its direction was never used. The letters A, B, C were only names, so that we could speak without ambiguity. Postulate 3 grants a circle with any centre and any distance; Postulate 1 grants a straight line from any point to any point; Definition 15 holds of every circle; Common Notion 1 holds of all things equal to the same thing. Therefore the same reasoning, word for word, applies to any finite straight line whatsoever. The conclusion is universal because the causes are universal.
Euclid closes: (Being) what it was required to do. In the Latin of the schools, quod erat faciendum. This is the closing formula of a problem, and it says: the thing demanded has been carried out. Had this been a theorem, asserting a truth about something already given rather than producing something new, it would have closed which was to be demonstrated — quod erat demonstrandum.
Now gather the causes, so that you hold the whole proposition in mind at once:
Given: one finite straight line, AB, and nothing else.
Sought: an equilateral triangle having AB for one of its sides.
Constructed: a circle with centre A through B; a circle with centre B through A; and, from a point C where these circles cut one another, the straight lines CA and CB.
Proved: CA equals AB, because both are drawn from the centre A to the boundary of one circle; CB equals AB, because both are drawn from the centre B to the boundary of the other; therefore CA equals CB, since things equal to the same thing are equal to one another; therefore all three sides are equal, and the triangle is equilateral.
By these causes: Postulate 3, Postulate 1, Definition 15, Common Notion 1, Definition 20.
You now know this, in the strict sense of the word. You know the cause — why the sides are equal — and you know that, the construction being granted, it could not be otherwise. This is not something you believe on my word or Euclid’s. It is something you see.
And the mark that a man truly knows a thing is that he can teach it. So test yourself: find another person, and without notes and without a book, explain to him what was given, what was sought, what was drawn and by what permission, and why the three sides must be equal. If you can bring him to see it as you see it, you know this proposition. If you cannot, return to the demonstration and find the step you were only remembering rather than understanding.
VII. REFLECTION
Its place within the Elements
This proposition is not one of the quiet steps. It is used at once and often.
Proposition I.2 — to place at a given point a straight line equal to a given straight line — begins by constructing an equilateral triangle on the line joining the given point to one end of the given line, and it could not be begun without I.1. And I.2 in turn is needed for I.3, and I.3 is needed constantly thereafter, for it is by I.3 that Euclid cuts off from a greater line a part equal to a lesser. So the whole of Euclid’s power to transfer and compare lengths — which is to say, a great part of Book I — traces back to this first proposition.
I.9, to bisect a given rectilineal angle, constructs an equilateral triangle in the course of the proof. I.11, to draw a straight line at right angles to a given straight line from a given point on it, does the same. Thus the two most basic operations of the geometer after drawing a line — dividing an angle in half, and setting up a perpendicular — both rest here.
The equilateral triangle itself returns as an object of study in Book IV, where Euclid inscribes one in a given circle, and it is the face of the first of the five regular solids constructed in Book XIII, the last book of the Elements. So the figure produced in the very first proposition is among the last things the work considers.
Its significance in philosophy
Proclus, in his commentary on this first book, uses this very proposition to set out the six parts of a complete proposition — enunciation, exposition, specification, construction, demonstration, conclusion — which are the six headings of this lesson. The form you have just followed is not a modern teaching device; it is the analysis the ancient commentators drew from Euclid’s own practice.
More important is what this proposition shows about the structure of demonstrative science, as Aristotle sets it out in the Posterior Analytics. He argues that not everything can be demonstrated: if every truth required a prior demonstration, we should either argue in a circle or go back without end, and in neither case would we ever know anything. Therefore demonstration must begin from principles that are not themselves demonstrated but are known in another way — definitions, and postulates, and axioms common to all reasoning. This first proposition of Euclid is the plainest exhibition of that doctrine in all of literature. Count what it rests on: two postulates, one definition, one common notion, and one more definition to name the result. Not one of these is proved, and not one of them needs to be. Everything after them is proved, and proved by them.
Notice also the role of definition. Definition 15 does not merely label the circle; it is the working premise of the entire demonstration. Aristotle held that a real definition states the essence of a thing, and that the essence is the cause of the thing’s properties. Here you can watch that happen: the equality of the sides is drawn out of what a circle is. And notice, on the other side, the discipline mentioned earlier — that defining the equilateral triangle in Definition 20 did not establish that one exists. To know what a thing is, and to know that it is, are two different acts of the mind. Euclid keeps them apart with great care, and the Scholastic doctors made much of this distinction.
Its significance in theology
St. Thomas Aquinas, at the very opening of the Summa Theologiae, asks whether sacred doctrine is a science, and answers by an appeal to the structure you have just used. Some sciences, he says, proceed from principles known by the natural light of the intellect, as arithmetic and geometry do; others proceed from principles known by the light of a higher science, as the science of optics proceeds from principles established in geometry. Sacred doctrine is of this second kind: it proceeds from the articles of faith, which it does not prove but receives, and from them it argues to conclusions. The comparison only holds because geometry visibly does what Aquinas says every science does — accept certain principles without proof, and then reason with strict necessity from them. A student who has worked through this proposition understands what Aquinas is claiming; a student who has not, does not.
There is a further point, and I state it modestly, as it should be stated. The geometer does not create the equality of CA and CB; he finds it. The necessity he meets is not of his making, and he cannot alter it by any wish. Whatever measure, order, and proportion he discovers in things, he discovers as already there. Christian doctrine holds that this order is in things because God put it there, having disposed all things in measure and number and weight. Geometry does not prove this; it is not the business of geometry to prove it. But the geometer’s daily experience of encountering an intelligible order he did not invent is one of the reasons why the study of mathematics has always been counted a preparation for the study of divine things.
Its descendants in modern mathematics
Several things you may have met in a modern classroom are this proposition, or its immediate parts, in other dress.
The construction itself survives untouched. What is taught today as “constructing an equilateral triangle with compass and straightedge,” and what is taught as “constructing a 60-degree angle,” are Euclid I.1 exactly. The 60-degree angle is obtained by drawing two arcs of equal radius from the ends of a segment and joining the crossing point; that is Step 1, Step 2, and Step 3 above, with the sides discarded and only the angle kept.
The reasoning is likewise preserved in analytic geometry. In modern coordinates, a circle whose centre is the point named by the pair a and b, and whose radius is r, is written as the equation
(x – a)^2 + (y – b)^2 = r^2
which is read: the square on the difference between x and a, added to the square on the difference between y and b, is equal to the square on r. That equation is nothing but Definition 15 restated in the language of coordinates: it is the condition that a point be at the fixed distance r from the fixed centre. And the modern exercise of solving two such equations at once, to find where two circles meet, is Step 3 of this construction. When you set up those two equations and conclude that the point found is at the distance r from both centres, you are repeating the first and second steps of this demonstration in algebraic notation.
Common Notion 1 has descendants everywhere. In modern algebra it appears as the transitive property of equality: if a is equal to c, and b is equal to c, then a is equal to b. Every time you substitute one expression for another because both are equal to a third, you are using what Euclid granted in his first common notion, and using it exactly as he used it in the fourth step above.
Finally, this triangle stands behind a fact you may already have memorized in trigonometry. Because the equilateral triangle has three equal sides and therefore three equal angles, each of its angles is a third of two right angles, that is, 60 degrees; and if such a triangle is divided in half by a perpendicular from one vertex, the half is the familiar right triangle with angles of 30 and 60 degrees, in which the side opposite the 30-degree angle is exactly half the hypotenuse. That is why the sine of 30 degrees is equal to one half — a value not measured or approximated, but demonstrated, and demonstrated ultimately from the figure constructed here. (The steps that fill in that account, concerning the angles of a triangle, come later in Book I; I mention the connection so that you may look for it, not as though it had been proved now.)
The end of the study
Consider what you now hold. Not a drawing on a page, which is smudged and imperfect and will be thrown away. You hold this: that upon any finite straight line whatsoever, in any place, of any length, an equilateral triangle can be constructed, and you know the cause, and you know it could not be otherwise. That was true before any geometer was born and before the earth was made. It will be true when this world has passed away. No authority can revise it, no vote can repeal it, no change in circumstances can touch it, and no one who understands the demonstration can doubt it.
This is what it is to know, in the strict and proper sense. Most of what men carry in their heads is not of this kind, and most men have never had the experience by which the difference can be recognized. You have now had it once. Guard the memory of it, and use it as a measure: when you are next told something in any subject whatever, ask whether it is held as you hold this, and if not, ask why not.
The appetite for this kind of knowing, once it has been awakened, is the spirit of philosophy. Euclid awakens it by giving it its first satisfaction. Go on now to Proposition 2, where you will find that this triangle is immediately put to work, and that the power to carry a length from one place to another — which we could not use in this lesson — is about to be won.
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