Return to QRV-321 Classical Geometry I
In this lesson, we will study Euclid, Book I, Proposition 2.
I. ENUNCIATION
The proposition we are now to contemplate is stated by Euclid in these words:
“To place at a given point (as an extremity) a straight line equal to a given straight line.”
Read the words slowly. We are given two things: a point somewhere, and a straight line somewhere else. We are required to produce a straight line which has the given point for one of its ends, and which is equal to the given straight line. The parenthesis, “as an extremity,” is Euclid’s own guard against a misunderstanding. An extremity of a line is one of its two ends, as Definition 3 says: “The extremities of a line are points.” So the new line must not merely pass through the given point, nor have the given point somewhere along its middle; the given point must be one of its two ends. The line must begin there.
Now, why do we take up such a task at all?
Geometry is a speculative science. That means it is not undertaken in order to make something or to get something done, but in order to know. Its objects — points, lines, angles, figures — are not found in a separate world reserved for them. They are found in the sensible things all around us, in the edge of a table and the rim of a wheel and the corner of a room. The geometer reaches them by abstraction: he considers the quantity and the shape of a thing apart from the wood or iron or air in which he always meets it. Knowledge begins in the senses; but what the geometer comes to hold is no longer a fact about any particular piece of wood. It is true always and everywhere, and it could not be otherwise. This is why Plato, in the seventh book of the Republic, says 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.
Within that end, this proposition has a particular and important office, and it is not merely the office of a handy trick.
Consider what a man ordinarily does when he wants a length in a new place. He takes a ruler, or a pair of dividers, and he carries the length over. That is measurement, and it rests on trust: trust in the instrument, trust in the steadiness of his hand, trust that the metal did not flex on the way. Whatever he obtains in that manner, he does not know with certitude; he estimates it. Euclid will not proceed in that manner. He has laid down at the beginning exactly what he is permitted to do, and nothing among his postulates permits him to pick up a length and set it down elsewhere. His third postulate says, “To describe a circle with any centre and distance.” As it is used throughout the Elements, this means that if we have a point, and a straight line already drawn from that point, we may describe the circle which has that point for centre and passes through the far end of that line. It does not permit us to take a length lying in one place and use it as the distance for a circle centred in another place. That would be to assume the very thing this proposition asks us to accomplish.
So the purpose of this proposition is this: to show that a length can be transferred from one place to another, not by trusting an instrument, but by demonstration — so that when we say the new line is equal to the old one, we say it because we have seen the cause of the equality and cannot doubt it. That is the whole difference between opinion and knowledge, and here, at the very second proposition of the Elements, the student is made to feel it. Afterwards, whenever anyone tells him that two things are equal, he has a standard by which to ask: is this demonstrated, or merely asserted?
Let us recall where we now stand.
Before anything was proved, we were given three kinds of principles. First, the definitions, which tell us what we are speaking about: what a point is, what a line is, what a straight line is, what a circle is, what its centre is, what an equilateral triangle is. Definitions do not assert that anything exists; they only fix the meaning of our words. Second, the postulates, which grant us the operations we may perform and one or two things we may assume: to draw a straight line from any point to any point; to produce a finite straight line continuously in a straight line; to describe a circle with any centre and distance; that all right angles are equal to one another; and the fifth postulate concerning lines that meet. Third, the common notions, which are principles of quantity in general and not of geometry alone: that things equal to the same thing are equal to one another; that if equals be added to equals the wholes are equal; that if equals be subtracted from equals the remainders are equal; that things which coincide are equal; and that the whole is greater than the part.
Then, in Proposition 1, we learned to construct an equilateral triangle upon a given finite straight line. That construction gave us something we shall now use: a point from which two equal straight lines run to the two ends of a given line. That is the only proposition we have. What remained wanting was any means of connecting one part of the plane with another — of making a length here answer to a length there. Proposition 1 gives us a figure standing on one line; Proposition 2 uses that figure to reach across to a distant point. This is why it comes next.
Finally, note what kind of proposition this is. In the Elements there are two kinds. A problem requires something to be constructed or done; it ends with the formula “(Being) what it was required to do.” A theorem asserts that something is true of what is given; it ends with “(Being) what it was required to prove.” This proposition, like the first, is a problem. It says “To place,” which is a thing to be done. So the certitude we are about to acquire is the certitude that a certain thing can be brought about by the operations we have been granted, and that what is brought about is truly what was demanded.
II. EXPOSITION
Now we set the general statement out in particular terms, and we name the particular things with letters.
We name things with letters so that we may speak of them precisely and without ambiguity. If we said “the point” and “the line” and “the other line,” we should soon lose ourselves. A point is named by a single letter placed at it. A straight line is named by the two letters at its extremities, so that the line whose ends are marked B and C is called BC, and it may equally be called CB, since it is the same line either way.
Euclid says: “Let A be the given point, and BC the given straight line.”
So we are given a point, and we call it A. We are given a straight line, finite in length, having two extremities, and we mark those extremities B and C, and call the line BC.
Now be exact about what is given, and refuse to grant yourself anything more.
We are given that A is a point. We are not told where it lies with respect to BC — whether near or far, above or below, to the right or to the left. Nothing in the demonstration will depend on any of that.
We are given that BC is a straight line, bounded at both ends. We are not told how long it is. Nothing will depend on its length.
We are not given any instrument for measuring. We are not given any number attached to BC. We are not given permission to move BC, nor to lay it against anything, nor to slide it. Whatever we do beyond what is given must be licensed by a definition, a postulate, a common notion, or by Proposition 1, which is all we have so far demonstrated.
One further remark on what is given. Euclid, in the construction, will join A to B. For this to be a joining of two points, A and B must be two points and not one. If it should happen that the given point A were the very extremity B of the given line, there would be nothing to do at all: the line BC already has A for an extremity, and it is equal to itself. So that case needs no construction, and Euclid passes it by. In everything that follows, then, understand A and B to be distinct points.
III. SPECIFICATION
Euclid states the goal in the particular terms just set out: “Thus it is required to place at the point A (as an extremity) a straight line equal to the given straight line BC.”
What is unknown, and what we seek, is this: a straight line, one of whose extremities is the point A, and which is equal to the straight line BC.
Observe carefully three things about this goal.
First, we are not asked to find where such a line is; we are asked to produce one, and to prove of the line we have produced that it is equal to BC. The proof is the substance of the work. A line drawn at hazard and pronounced equal would be worth nothing.
Second, we are not told in what direction the new line is to run from A. That is left free. This is important: the problem asks for a straight line equal to BC and having A for an end, and any one such line satisfies it. Our construction will happen to give us one particular direction, determined by the figure we build; that is no defect, because one is what was asked.
Third, the whole difficulty lies in the distance between what we hold and what we seek. We hold a length, BC, at one place, and a point, A, at another place, with nothing between them. We have no permission to carry the length across. We have permission to draw straight lines between points, to prolong straight lines, and to describe circles about a centre through a point already joined to that centre. Out of those permissions alone, and out of the common notions about equals, the crossing must be made. The construction is the bridge; the demonstration shows that the bridge holds.
IV. CONSTRUCTION
This proposition is a problem, and therefore it has a construction, and the construction is elaborate. Take it one step at a time. For each step I shall say what is done, what permits it, and what we have gained.
Step one. Euclid says: “From the point A to the point B let the straight line AB be joined.”
To join two points means to draw the straight line which has those two points for its extremities. What permits it is Postulate 1: “To draw a straight line from any point to any point.” Notice that the postulate says any point to any point; it does not require the points to be near, or conveniently placed. A is a point and B is a point, and they are distinct; so the straight line AB may be drawn.
Stated plainly: draw the straight line from the given point A to the end B of the given line.
What have we gained? A single finite straight line, AB, which has one end at the given point and the other end at an extremity of the given line. This is the first link between the two places.
Step two. Euclid says: “and on it let the equilateral triangle DAB be constructed.”
An equilateral triangle, by Definition 20, is a trilateral figure which has its three sides equal. What permits us to construct one upon a given finite straight line is Proposition 1, which we have already demonstrated: on any given finite straight line an equilateral triangle can be constructed. Here the given finite straight line is AB, which we have just drawn.
The triangle is named by the three letters at its three angular points: D, A, and B. Two of these we already have; the third, D, is the new point which Proposition 1 supplies, the point at which the two circles of that construction cut one another. The three sides of the triangle are DA, AB, and DB, and by Definition 20 together with Proposition 1 these three straight lines are equal to one another.
Stated plainly: on the line AB build an equilateral triangle, and call its new corner D.
What have we gained? Precisely this: a point D from which two equal straight lines, DA and DB, run out, one to the point A and one to the point B. That equality, DA equal to DB, is the hinge of the whole proposition. Keep hold of it.
The triangle may be constructed on either side of AB; Proposition 1 permits either. Nothing that follows depends on which side is chosen.
Step three. Euclid says: “Let the straight lines AE, BF be produced in a straight line with DA, DB.”
To produce a straight line means to prolong it beyond its end, continuing in the same straight direction. What permits it is Postulate 2: “To produce a finite straight line continuously in a straight line.” We prolong DA beyond the point A, and we prolong DB beyond the point B, as far as we shall need.
Euclid’s manner of naming here is worth a moment. He does not name the whole prolonged lines; he names the added portions. The prolongation of DA beyond A he calls AE, so that D, A, and E lie in one straight line, in that order. The prolongation of DB beyond B he calls BF, so that D, B, and F lie in one straight line, in that order. When afterwards we speak of “the straight line DE” we mean the whole line from D through A to E; and by “the straight line DF” the whole line from D through B to F.
Stated plainly: prolong the two equal sides of the triangle, each beyond the end that is not D.
What have we gained? Two straight roads leading out from D, one passing through A, the other passing through B, and each able to be continued as far as we please. Since DA and DB are equal, the two roads reach the points A and B at the same distance from D.
Step four. Euclid says: “with centre B and distance BC let the circle CGH be described.”
Recall Definition 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.” And Definition 16: “And the point is called the centre of the circle.” So the centre of a circle is that interior point from which all straight lines drawn to the circumference are equal.
What permits the description of a circle is Postulate 3: “To describe a circle with any centre and distance.” Here the centre is B, and the distance is BC — that is, the straight line BC, which has B for one of its extremities and is therefore already a line drawn from the centre we have chosen. This is exactly the use of the postulate which is allowed us: we describe the circle about B which passes through C. We are not helping ourselves to any forbidden transfer of length. The length BC is used where it already lies, at B.
The circle is named CGH by three of the points on its circumference. A circle is named in this way so that we may distinguish it from other circles in the figure by naming points which lie upon it. C is one such point, since the circle passes through C. H is simply another mark on the circumference, of no further use. G is the point that matters: it is the point in which this circle cuts the straight line BF — that is, the prolongation of DB beyond B.
Stated plainly: about the point B, as centre, describe the circle which passes through C; and let G be the point where that circle crosses the prolonged side DB.
Here I must supply something that Euclid does not say, and tell you that I am supplying it. Euclid takes it for granted that the circle about B does cross the straight line BF, and so that the point G exists. Why should we believe it? Because B is the centre of the circle, and therefore B lies within it; and the straight line BF goes out from B and may be produced as far as we please by Postulate 2, while every point of the circumference is at the fixed distance BC from B. A line which starts inside the circle and runs out beyond every fixed distance must somewhere pass through the circumference. This is evident to the imagination, and no one doubts it; but it is not written among Euclid’s postulates. It is what is called an assumption of continuity — the assumption that a line has no gaps in it, so that a line passing from inside a circle to outside it must meet the circle. Later mathematicians, seeing that Euclid used this silently both here and in Proposition 1, wrote it out as an explicit principle. I point this out not to shake your confidence, but for the opposite reason: a student of demonstration should know exactly which of his steps rest on stated principles and which rest on principles that were used before they were stated. That is part of learning what certitude is.
What have we gained? A point G on the straight line drawn from D through B, such that BG is equal to BC. We have not yet said this; it belongs to the demonstration. But this is the point of the step: the length of BC has been laid out along the road DBF, beginning at B.
Step five. Euclid says: “and again, with centre D and distance DG let the circle GKL be described.”
Again Postulate 3 permits it, and again the permission is exactly satisfied. The centre is D. The distance is DG. Is DG a straight line drawn from D? It is: for D, B, and G all lie upon the one straight line DBF, in that order, and so DG is a portion of that straight line, having D for one extremity and G for the other. Therefore we may describe about D the circle which passes through G.
This circle is named GKL. It passes through G, as we have just said. K is another mark on the circumference, of no further use. L is the point that matters: it is the point in which this circle cuts the straight line DE — that is, the other road out from D, the prolongation of DA beyond A.
Stated plainly: about the point D, as centre, describe the circle which passes through G; and let L be the point where that circle crosses the other prolonged side, the one through A.
Two things must be supplied here, and I shall supply them plainly.
First, that this circle does cut the straight line DE, so that the point L exists. The reason is the same as before: D is the centre and so lies within the circle, and the straight line DE goes out from D and may be produced as far as we please, while the circumference lies at the fixed distance DG from D. This is again the assumption of continuity.
Second, and this is needed for the demonstration, that L falls beyond A — that is, that the order of the three points on the line is D, then A, then L. Here is the reason. The point B lies between D and G, since G was taken on the prolongation of DB beyond B. Therefore DB is a part of DG, and by Common Notion 5, “the whole is greater than the part,” DG is greater than DB. But DB is equal to DA, since they are two sides of the equilateral triangle DAB. Therefore DG is greater than DA. And DL is a straight line drawn from the centre D to the circumference, as DG is, so DL is equal to DG, by Definition 15; hence DL also is greater than DA. Now DL and DA are measured out from the same point D along the same straight line DE. Since DL is the greater, the point A, which is nearer to D, must lie between D and L. Euclid does not stop to say this, but the demonstration will speak of “the remainder AL,” and one may only speak of a remainder when there is a whole and a part taken away from it. So the order of the points had to be established, and now it is.
What have we gained? A point L upon the straight line drawn from D through A, beyond A, such that DL is equal to DG. And AL, the portion of that line between A and L, is the straight line which will prove to be the one we sought.
Now, the whole construction gathered into a few sentences, so that you may hold it at once: join the given point A to the end B of the given line; on AB build an equilateral triangle with new vertex D; prolong the two sides DA and DB beyond A and beyond B; about B describe the circle through C, cutting the prolongation of DB at G; about D describe the circle through G, cutting the prolongation of DA at L. The line AL is the line sought.
You may now take a straight edge and a compass and draw this figure with your own hand. Do so only now, after you have followed the words, and understand what you are doing when you do it: you are not learning the construction from the drawing, you are exhibiting a construction you already understand. And remember, when you look at what you have drawn, that it is not the geometrical thing. A geometrical line has no breadth and cannot be seen; the line you draw has breadth, and is seen. The drawing helps the imagination and at the same time misrepresents what it aids. Reason holds the line; the eye never does.
V. DEMONSTRATION
We now prove that the line AL, which the construction has given us, is equal to the given line BC. Nothing in this proof will appeal to how the figure looks. Every step will rest on a definition, a postulate, a common notion, or Proposition 1.
First step: BC is equal to BG.
Euclid says: “Then, since the point B is the centre of the circle CGH, BC is equal to BG.”
The premises are these. By the construction, B is the centre of the circle CGH, and C is a point on its circumference, since the circle was described through C; and G is a point on its circumference, since G was taken as the point where that circle cuts the line BF. And by Definition 15, a circle is such that “all the straight lines falling upon it from one point among those lying within the figure are equal to one another,” that one point being, by Definition 16, the centre.
The reasoning joins them thus: all straight lines drawn from the centre of a circle to its circumference are equal to one another; BC and BG are straight lines drawn from the centre B of this circle to its circumference; therefore BC is equal to BG.
Could this be otherwise? It could not, unless the word “circle” meant something other than what Definition 15 says. The equality of the lines from the centre is not a discovery about circles; it is what makes a figure a circle. To grant that CGH is a circle with centre B, and to deny that BC is equal to BG, is to contradict oneself.
What has now been established: the given length BC has an equal, BG, lying along the straight line that runs from D through B. Notice what this gains us. We could not bring BC to A. But we have made a copy of BC in a place where it is joined onto something equal to something at A.
Second step: DL is equal to DG.
Euclid says: “Again, since the point D is the centre of the circle GKL, DL is equal to DG.”
The premises are of the very same kind. By the construction, D is the centre of the circle GKL; G lies on its circumference, since the circle was described through G; and L lies on its circumference, since L was taken where the circle cuts the line DE. And by Definitions 15 and 16, all straight lines from the centre to the circumference are equal.
Therefore DL is equal to DG. Again this could not be otherwise, for the same reason as before.
What has now been established: two whole lines out of D, namely DL along one road and DG along the other, are equal to one another.
Third step: DA is equal to DB.
Euclid says: “And in these DA is equal to DB.”
Where does this come from? From the construction, step two. The triangle DAB was constructed by Proposition 1 to be equilateral, and by Definition 20 an equilateral triangle “has its three sides equal.” The three sides are DA, AB, and DB. Therefore DA is equal to DB.
Notice the phrase “and in these.” Euclid means: within those two equal wholes, DL and DG, there are these two equal parts, DA and DB. For DA is a part of DL, since the order of points on that line is D, A, L; and DB is a part of DG, since the order of points on that line is D, B, G. Both of these facts about order were established in the construction, and that is why we took the trouble to establish them.
Fourth step: AL is equal to BG.
Euclid says: “therefore the remainder AL is equal to the remainder BG.”
The premises are: the whole DL is equal to the whole DG (second step); the part DA taken from the first is equal to the part DB taken from the second (third step). And Common Notion 3 says: “If equals be subtracted from equals, the remainders are equal.”
Now apply it. From the equal wholes DL and DG, subtract the equal parts DA and DB. What remains of DL, after DA is taken away, is AL. What remains of DG, after DB is taken away, is BG. Therefore AL is equal to BG.
Dwell on this step, for it is the crossing of the distance. The equality of AL and BG has not been seen, nor measured, nor guessed. It follows by a principle about quantity in general — that equals taken from equals leave equals — which we would grant of weights or of times or of numbers as readily as of lines. Grant the common notion, grant the two equalities, and the conclusion cannot be refused. There is no room left in which to doubt it.
Note also that Common Notion 3 could not have been applied unless DA were truly a part of DL and DB truly a part of DG. That is why the construction had to establish the order of the points, and why I supplied the proof that L lies beyond A. A step of subtraction that is not licensed by a genuine whole and part is no step at all.
Fifth step: each of AL and BC is equal to BG.
Euclid says: “But BC was also proved equal to BG; therefore each of the straight lines AL, BC is equal to BG.”
This is only the gathering together of what we already hold. From the first step, BC is equal to BG. From the fourth step, AL is equal to BG. So there is one line, BG, to which each of the two lines AL and BC is equal.
Sixth step: AL is equal to BC.
Euclid says: “And things which are equal to the same thing are also equal to one another; therefore AL is also equal to BC.”
The premise supplied here is Common Notion 1: “Things which are equal to the same thing are also equal to one another.” The same thing is BG. The two things equal to it are AL and BC. Therefore AL is equal to BC.
Look closely at the shape of this reasoning, because it is the shape of demonstration itself. We wished to join two extremes, AL and BC, which had nothing directly in common: one lies at the given point, the other is the given line, and they are in different parts of the plane. We joined them through a third thing, BG, which is related to each. In the study of reasoning, such a third thing is called the middle term, and every demonstration proceeds by finding one. The whole art of the construction was the finding of a suitable middle: BG was chosen because it could be proved equal to BC by the circle about B, and equal to AL by the circle about D together with the equal sides of the equilateral triangle.
Seventh step: the thing required has been done.
The line AL is a straight line, being a portion of the straight line DE. One of its extremities is the point A — the given point. And it has been proved equal to the given straight line BC. Therefore at the given point A there has been placed, as an extremity, a straight line equal to the given straight line BC.
Nothing remains. There is no gap in the chain. Every link is either a definition of a term we ourselves fixed, a postulate we were granted, a common notion about quantity, or Proposition 1, which was itself demonstrated from those same principles — with the single assumption of continuity, which I named openly where it was used.
VI. CONCLUSION
Return now to the universal statement with which we began: to place at a given point, as an extremity, a straight line equal to a given straight line. This now stands proved — not for the particular point and the particular line we lettered A and BC, but for any point whatever and any bounded straight line whatever.
Why may we say so? Because nothing particular to our figure was used. We never used the length of BC; we never used the position of A, nor how far it was from B, nor on which side it lay. We never used which side of AB the triangle DAB was built on. Every step appealed only to the definition of a circle, the definition of an equilateral triangle, the postulates of drawing, producing, and describing, and the common notions about equals. Whatever point and whatever line another man may be given, he may do exactly what we did, and the same reasoning will hold of his figure word for word. That is what it means for a geometrical conclusion to be universal.
Euclid closes: “Therefore at the given point A the straight line AL is placed equal to the given straight line BC. (Being) what it was required to do.”
That closing formula belongs to a problem. A problem demands that something be constructed or done, and when it has been done and proved to be what was demanded, we say: which was to be done. A theorem, by contrast, asserts something true of what is given, and closes: which was to be demonstrated. This lesson has given us a problem, and therefore a power: we can now do something we could not do before.
Let us gather the causes, so that you hold the whole proposition in mind at once.
This was given: a point A, and a bounded straight line BC.
This was sought: a straight line having A for an extremity and equal to BC.
This was constructed: the line AB joining the point to the line’s end; the equilateral triangle DAB upon it, giving a point D with DA equal to DB; the prolongation of DA and DB beyond A and beyond B; the circle about B through C, cutting the prolongation of DB at G; the circle about D through G, cutting the prolongation of DA at L.
This was proved: BC equal to BG, because both are drawn from the centre B to the circumference of one circle; DL equal to DG, because both are drawn from the centre D to the circumference of one circle; DA equal to DB, because they are sides of an equilateral triangle; therefore, equals being taken from equals, AL equal to BG; and therefore, things equal to the same thing being equal to one another, AL equal to BC.
You now know this, in the strict sense of knowing. You know it not because a book said it and not because a drawing looked convincing, but because you have seen the cause, and have seen that the conclusion cannot be otherwise while the principles stand. And the sure mark that a man knows a thing is that he can teach it. So take this test: go to another person, without notes, and explain to him why AL must be equal to BC. If you can lead him from the given point and the given line, through the triangle and the two circles, to the equality, naming at each step what permits it, then you know Proposition 2. If you stumble, the stumble will show you exactly which step you have not yet made your own, and you should return to it.
VII. REFLECTION
Let us consider honestly what this proposition is worth.
Within the Elements, its immediate office is to make Proposition 3 possible. Proposition 3 says: given two unequal straight lines, to cut off from the greater a straight line equal to the less. Its very first move is to place at an extremity of the greater line a straight line equal to the less — which is what we have just learned to do. And Proposition 3 is used constantly thereafter: in Proposition 5 concerning the angles of an isosceles triangle, in Proposition 11 and Proposition 16 and Proposition 22, and throughout the later books, whenever a length must be marked off equal to another length. So Proposition 2 is a foundation stone which afterwards is rarely mentioned by name, because the propositions that rest on it stand between it and the rest of the work. That is the ordinary fate of foundations. It is not one of the great and famous theorems; it is one of the few small propositions without which the great ones could not be reached.
Its deeper significance lies in Euclid’s refusal to take a shortcut. Any craftsman would have said: set your dividers to BC, carry them to A, and strike the length off there. Euclid does not permit it, because he has not granted himself the operation. This is the discipline that makes the Elements a science rather than a manual: whatever is not among the principles must be demonstrated, and what is demonstrated is then known with the certitude of the principles themselves. Aristotle, in the Posterior Analytics, teaches that there must be first principles which are not themselves demonstrated — for otherwise nothing could ever be proved, since the proofs would never end — and that everything else must be reduced to them. Euclid’s postulates are such principles: we do not prove that a straight line may be drawn from any point to any point; we grant it. But the transfer of a length is not such a principle, and so it is proved. A student who grasps the difference between what is granted and what is proved has learned something he will use in every subject for the rest of his life. Most men never learn it, and so cannot tell knowledge from opinion.
The proposition also displays plainly the structure of demonstration by a middle term, which I pointed out in the fifth and sixth steps. Two things which cannot be compared directly are compared through a third to which both are related. Common Notion 1 is the principle of that comparison. In modern arithmetic and algebra the same principle is called the transitivity of equality, and it is written: if a equals c and b equals c, then a equals b. Every time a student in algebra substitutes one expression for another because both are equal to a third, he is using Euclid’s first common notion, and Proposition 2 is one of the earliest and clearest instances of its use in a demonstration.
There is a further descendant worth naming exactly. In modern geometrical writing, a distinction is sometimes drawn between a “collapsing” compass, which cannot hold its opening when lifted, and a rigid compass, which can. Euclid’s third postulate, as he uses it, gives him only the first: he may describe a circle about a centre through a point already joined to that centre. What Proposition 2 proves is that this weaker instrument can do everything the stronger one can do, since any length can be reproduced at any point by construction alone. Modern textbooks call this the compass equivalence theorem, and this proposition is its original proof. That is a real and exact descent, not a loose comparison.
We may add one more honest observation. Modern treatments of geometry often take as an axiom that lengths may be carried about the plane without change — that is, they assume the existence of rigid motions, or they assume outright that a segment of any given length can be laid off from any point in any direction. Euclid assumes no such thing here; he derives it. Whether it is better to assume it or to derive it is a question about the arrangement of a science, and different arrangements have been given by good men. But the student should see that in Euclid it is a conclusion, and that he now possesses the reason for it, whereas one who is merely handed the axiom possesses only the permission.
And we should not forget what was said in the construction about continuity: Euclid used, without stating, the principle that a line passing from inside a circle to outside it must meet the circle. That silence was noticed by later commentators and was repaired in the nineteenth century, when the principles of continuity were set down explicitly. This is worth knowing because it shows that even the most rigorous work of antiquity had a hidden premise, and that finding hidden premises is itself part of the growth of a science. It does not weaken what we have proved; it locates precisely what our proof depends on.
In theology, the Scholastic doctors used the certitude of demonstrated conclusions as the very model for the order of knowledge. Saint Thomas Aquinas, at the beginning of the Summa Theologiae, explains that a science may take its principles from a higher science and then proceed to conclusions with certainty, as the practitioners of some arts take on faith the principles established by others; and sacred doctrine, he says, takes its principles from God’s own knowledge and proceeds from them. The geometer’s experience of drawing a necessary conclusion from principles he does not prove is the natural image of that order. Furthermore, the order and measure which the geometer finds in things is not of his own making. He finds it; he does not invent it. And the ascent of the mind to God runs through creatures rather than away from them: the necessity and the measure discovered in the things God made are a real participation in the truth of the Eternal.
Consider now what you hold. That a straight line equal to any given straight line can be set at any given point, and that AL must be equal to BC when the construction has been made, was true before you were born, and will be true when this world has passed away. It does not depend on any authority, and no opinion, no vote, no change of fashion or of circumstance can touch it. You did not receive it; you saw it. This is what it is to know. And when a man has once tasted knowing of this kind, he becomes unwilling to accept less in other matters, and begins to ask everywhere: is this demonstrated, or merely asserted? That unwillingness, that appetite, is the spirit of philosophy which Plato said geometry creates in the soul.
Go now to Proposition 3, where we shall use this new power to cut off from a greater straight line a part equal to a lesser.
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