CLAA – Eucild's Elements - Book I, Common Notions or Axioms

QRV-321 Lesson 003 – Book I, Common Notions

I. INTRODUCTION

This lesson studies the five Common Notions placed at the head of Book I of Euclid’s Elements of Geometry.

Every science is built out of two kinds of statements: those that are proved, and those from which the proofs are made. The statements that are proved are called propositions, theorems and problems. The statements from which proofs are made are called principles. A science cannot prove everything, because every proof must start from something already granted; if nothing were granted, nothing could ever be shown. So at the beginning of a science the principles are set down openly, and the reader is asked to accept them before any demonstration begins.

Euclid sets down three kinds of principles at the beginning of Book I, in this order: Definitions, Postulates, and Common Notions. The Definitions tell you what the words mean — what a point is, what a line is, what a circle is. The Postulates ask you to grant certain things that belong to geometry alone, chiefly that certain constructions can be made: that a straight line can be drawn from any point to any point, that a circle can be described with any centre and distance, and the like. The Common Notions are the third kind. They are not about the meanings of words, and they are not requests to grant a construction. They are true statements about quantity in general — about equals, wholes and parts — which are used in the reasoning of geometry but are not peculiar to geometry. They are equally true of numbers, of weights, of times and of lines. That is why they are called common: they are common to more than one science, and they are known in common by all men who understand the words.

Euclid of Alexandria composed the Elements about 300 years before Christ, in Alexandria in Egypt, in the reign of Ptolemy I. He did not invent all the matter of the work. He gathered the geometry of the Pythagoreans, of Eudoxus, of Theaetetus and of others, and arranged the whole in demonstrative order, so that nothing is used before it has been proved, and everything rests at last upon the definitions, postulates and common notions of Book I. Because of this order the work became, and remained for more than two thousand years, the model of a demonstrative science.

The Greek word Euclid uses is koinai ennoiai, common notions. Aristotle, writing a generation before Euclid, treats the same principles in the Posterior Analytics and calls them axioms, or common principles, and gives as his own example the very rule that if equals be taken from equals, equals remain. Boethius, the Roman senator and philosopher who died about the year 524 and who translated much of Greek learning into Latin for the West, renders the phrase as communes animi conceptiones, common conceptions of the mind, and defines such a statement thus: a common conception of the mind is a statement which anyone approves as soon as he has heard it. Saint Thomas Aquinas, commenting on Boethius and again in the Summa Theologiae, takes “the whole is greater than the part” as his standing example of a proposition self-evident to everyone, because everyone knows what a whole is and what a part is.

The text came down to us in Greek manuscripts, chiefly through the edition prepared by Theon of Alexandria in the fourth century after Christ; it was expounded in the fifth century by Proclus, whose Commentary on the First Book of Euclid’s Elements is our best ancient witness to how these principles were understood and disputed; it passed into Arabic, and thence back into Latin in the twelfth century by Adelard of Bath and later by Campanus of Novara; it was first printed at Venice in 1482 and first Englished by Henry Billingsley in 1570. Proclus records that some ancients wished to add further common notions, and that Apollonius of Perga attempted to demonstrate them. The tradition rejected both attempts: the additions because they are not needed, and the demonstrations because a first principle, by its nature, cannot be demonstrated without begging the question or falling into an endless chain of proofs. The five notions given here are what the tradition has kept.

Objectives of this lesson. When you have finished it you will be able to:

State what a common notion is, and distinguish it from a definition and from a postulate.
Recite the five Common Notions of Book I exactly.
Explain the meaning of each of the five, defining the terms equal, magnitude, whole, part, and coincide.
Give a worked example of the use of each common notion, in geometry or in number.
Explain why first principles are not demonstrated, and why this is not a defect in the science.

II. EXPOSITION

First, the terms that run through all five.

Magnitude means continuous quantity: a line, a surface, a solid, an angle. Continuous means that its parts join at a common boundary, so that it can be divided without end and has no smallest part. Number, by contrast, is discrete quantity: it is made of units that do not join. The common notions hold of both, and this is precisely why they are called common; but in Book I they are applied to magnitudes.

Equal, in Euclid, means equal in quantity — neither greater nor less. It does not mean the same thing, and it does not mean the same in shape. Two lines are equal when neither is longer; two angles are equal when neither is wider; two figures are equal when neither contains more surface. A triangle and a square may be equal, though nothing about them looks alike, if the surface contained in the one is neither more nor less than the surface contained in the other. Keep this firmly: equal is a comparison of quantity, and quantity only.

Further, only magnitudes of the same kind can be called equal or unequal. You may ask whether one line is equal to another line, or one angle to another angle, or one surface to another surface. You may not ask whether a line is equal to an angle. The question has no meaning, because they are not quantities of one kind. When the common notions speak of equals added to equals, they mean equals of the same kind added to equals of the same kind.

Whole and part. A part is a magnitude contained within another; the whole is that which contains it together with what remains. If a straight line AB is divided at C, then AC and CB are parts, and AB is the whole. If a triangle is divided by a line drawn from one vertex, the two triangles made are parts, and the first triangle is the whole.

Now take the five in Euclid’s order.

COMMON NOTION 1. Things which are equal to the same thing are also equal to one another.

Plainly: if a first magnitude is equal to a third, and a second magnitude is also equal to that same third, then the first and second are equal to each other. Let A, B and C be three magnitudes of one kind. If A is equal to C, and B is equal to C, then A is equal to B.

The reason is contained in what equality is. To say A is equal to C is to say A is neither greater nor less than C. To say B is equal to C is to say the same of B. Then A and B stand to C in exactly the same way in quantity, and so neither can exceed the other; for if A were greater than B, then, being equal to C, B would be less than C, contrary to what was granted.

Euclid uses this notion in the very first proposition of Book I, where an equilateral triangle is constructed on a given straight line AB. Two circles are drawn, one with centre A and distance AB, the other with centre B and distance BA, meeting at C. The lines are joined. Then, since A is the centre of the first circle, AC is equal to AB, for all straight lines drawn from the centre of a circle to its circumference are equal. Since B is the centre of the second circle, BC is equal to BA. So CA is equal to AB, and CB is equal to AB. And here the common notion is applied: things which are equal to the same thing are also equal to one another; therefore CA is equal to CB. The three sides are then all equal, and the triangle is equilateral. Without the common notion the two separate equalities could not be joined into the one conclusion needed.

The same holds in number. Seven and five make twelve; three times four make twelve; therefore seven and five are equal to three times four. In the language of modern algebra this is called the transitive property of equality; it is Euclid’s first common notion under another name.

COMMON NOTION 2. If equals be added to equals, the wholes are equal.

Plainly: take two equal magnitudes; add to the first some magnitude, and add to the second a magnitude equal to it; then the two sums are equal. If A is equal to B, and C is equal to D, then A together with C is equal to B together with D.

Notice that the addition must be of equals to equals. If you add a greater to one and a less to the other, the wholes will not be equal but unequal, and that is a different statement, which Euclid does not need here and does not set down.

A worked example from Euclid’s Book I, in the proof of Proposition 47, the theorem of the square on the hypotenuse. At a certain step it has been shown that the angle DBC is equal to the angle FBA, both being right angles. Then Euclid says: let the angle ABC be added to each; therefore the whole angle DBA is equal to the whole angle FBC. Equal angles were added to equal angles, and the wholes are equal. That is common notion 2, applied to angles.

A simple example in lines: let AB be equal to CD, and let BE be equal to DF, the parts lying in a straight line so that AE is the whole of AB and BE, and CF is the whole of CD and DF. Then AE is equal to CF.

In number: if two boys have equal sums, and each is given the same amount again, their sums remain equal. Modern textbooks call this the addition property of equality.

COMMON NOTION 3. If equals be subtracted from equals, the remainders are equal.

Plainly: take two equal magnitudes; take away from the first some magnitude, and take away from the second a magnitude equal to it; then what is left of the first is equal to what is left of the second. If A is equal to B, and C, a part of A, is equal to D, a part of B, then the remainder of A is equal to the remainder of B.

Euclid uses this in Proposition 5 of Book I, the theorem that the angles at the base of an isosceles triangle are equal. In the course of that proof two longer lines have been made, AF and AG, and shown to be equal; and within them the two equal sides AB and AC of the isosceles triangle lie. Euclid then says: since the whole AF is equal to the whole AG, and in these AB is equal to AC, the remainder BF is equal to the remainder CG. Equals were taken from equals, and the remainders are equal.

In number: twelve is equal to twelve; take five from the first and five from the second; seven remains from each, and the remainders are equal. Modern textbooks call this the subtraction property of equality.

Common notions 2 and 3 belong together. The second governs what happens when quantity is joined, the third what happens when quantity is separated. Together they are the reason why one may argue over the parts of a figure at all — why the pieces of a demonstration can be added up and taken away without the equalities collapsing.

COMMON NOTION 4. Things which coincide with one another are equal to one another.

Coincide translates the Greek word which means to fit upon, to be applied to and match exactly. Two magnitudes coincide when, the one being laid upon the other, the boundaries of the one fall upon the boundaries of the other, with nothing of either left over and nothing of either falling outside. This laying of one figure upon another is called superposition.

The notion says: if two magnitudes so fit upon each other, they are equal in quantity. The reason is evident from what equality is. If neither has anything outside the other, then neither is greater than the other, and so neither is less; therefore they are equal.

The chief use is in Proposition 4 of Book I, which proves that two triangles are equal in every respect if two sides and the included angle of the one are equal to two sides and the included angle of the other. Take triangles ABC and DEF, with AB equal to DE, AC equal to DF, and the angle at A equal to the angle at D. Apply the triangle ABC to the triangle DEF, placing the point A on the point D and the straight line AB along the straight line DE. Then, since AB is equal to DE, the point B falls on the point E. Since AB falls on DE and the angle at A is equal to the angle at D, the straight line AC falls along the straight line DF. Since AC is equal to DF, the point C falls on the point F. But B has fallen on E and C on F; therefore the base BC coincides with the base EF, for otherwise two straight lines would enclose a space, which is impossible. Since BC coincides with EF, BC is equal to EF, by the common notion; and the whole triangle coincides with the whole triangle, and is therefore equal to it, and the remaining angles coincide with the remaining angles and are equal to them.

Two cautions must be given, because they are commonly missed.

First, this common notion is stated in one direction only. It says that whatever coincides is equal. It does not say that whatever is equal coincides. And the converse is false: a triangle and a square may be equal in surface without coinciding at all, since they cannot be made to fit upon each other. Coincidence is a sufficient sign of equality, not a necessary one.

Second, the notion says nothing about how a figure is moved for the trial. It only tells you what follows if the figures do fit. Euclid uses superposition very sparingly, only where he cannot avoid it, and this restraint is itself a mark of his care. Proclus records that later geometers found the method uncomfortable and wished it were used less; but no one denied the truth of the notion itself, which is plain as soon as it is heard.

COMMON NOTION 5. The whole is greater than the part.

Plainly: any magnitude is greater than any magnitude contained within it, taken with something remaining outside. Greater means exceeding in quantity. The whole exceeds the part by exactly the rest of itself.

This is the standing example, in Aristotle and in Saint Thomas, of a proposition self-evident to all men. Its terms — whole, part, greater — are known to everyone who can speak. As soon as the sentence is heard and its terms are understood, the mind assents; and it cannot be proved by anything better known, because nothing is better known.

Its chief use in Book I is in proofs by contradiction. Euclid supposes the opposite of what he intends to prove, draws out of that supposition the consequence that some whole is equal to a part of itself, and then rejects the supposition as impossible.

So in Proposition 6, the converse of the isosceles theorem: if two angles of a triangle are equal to one another, the sides opposite them are also equal. Let the triangle be ABC with the angle ABC equal to the angle ACB. Suppose the sides AB and AC are not equal; then one is greater; let AB be the greater. Cut off from AB a part DB equal to AC, and join DC. Then, by Proposition 4, the triangle DBC is equal to the triangle ACB — the less to the greater, which is absurd. It is absurd precisely by this fifth common notion: the triangle DBC is a part of the triangle ACB, and the whole is greater than the part, so the two cannot be equal. Therefore AB is not unequal to AC; therefore it is equal.

Note also that the fifth common notion is what makes the words greater and less usable in the science at all. The first four concern equality; the fifth introduces inequality, and grounds it in the relation of whole and part.

A NOTE ON THE FORM OF THESE PRINCIPLES

You may ask: why does Euclid not prove these five statements? The answer belongs to logic and is given by Aristotle in the Posterior Analytics. A demonstration is a syllogism that produces knowledge, and its premises must be true, primary, immediate, better known than the conclusion, and causes of the conclusion. If every premise had itself to be demonstrated, then either the proofs would run back without end, so that nothing would ever be established, or they would circle round and prove a thing by itself, which proves nothing. Therefore there must be premises that are true and known without demonstration. These are the first principles. The common notions are of this kind. They are not accepted because Euclid says so, and not accepted as guesses; they are accepted because, the terms being understood, the mind sees that they are true and cannot be otherwise.

Saint Thomas distinguishes two sorts of self-evident propositions: those whose terms are known to all men, and which are therefore self-evident to all; and those whose terms are known only to the learned, and which are therefore self-evident only to them. The common notions of Euclid are of the first sort. This is exactly why they may be placed at the beginning of the work, before the reader has learned anything, and why the beginner can grant them honestly.

One last distinction, so that the three kinds of principle are not confused.

A definition says what a word means. “A point is that which has no part.” Nothing is asserted to exist; the meaning of a term is fixed.

A postulate asks that something be granted which belongs to this science in particular, and chiefly that a certain construction may be made. “Let it be granted that a straight line may be drawn from any point to any point.” It is peculiar to geometry, and it is a demand made of the learner.

A common notion states a truth about quantity as such, which the learner already knows and needs no demand to accept, and which is used in geometry but is not owned by geometry alone.

III. REFLECTION

Consider first what part of the soul this lesson perfects.

Aristotle distinguishes, in the sixth book of the Nicomachean Ethics, the habit by which we know first principles from the habit by which we know conclusions. The habit of conclusions is science, scientia: the settled ability to demonstrate a thing from its causes. The habit of first principles is understanding, intellectus, the Greek nous: the settled grasp of truths that are not demonstrated but seen. Science depends on understanding, because the premises of every demonstration must come from somewhere. A man who studies the common notions is not gathering conclusions; he is examining and steadying the principles from which all the conclusions of geometry will be drawn. He is exercising and perfecting his intellectus.

Consider next how these principles are known. They are not remembered from another life, and they are not looked at in some separate world of forms. Every man learns them by abstraction from the things his senses report. A child divides bread and sees that the loaf is more than the slice; he takes the same amount from two equal heaps and sees that the heaps remain equal; he lays one stick upon another and sees that they match. From many such sensible cases the intellect abstracts the terms whole, part, equal, greater, and then, holding the terms in view, sees that the propositions are true universally, of all magnitudes whatsoever, and not merely of the loaves and sticks. This is the ordinary road of human knowledge: all our knowing begins in the senses and proceeds by abstraction. Geometry stands at the second degree of remove from matter — it considers quantity apart from the particular sensible matter in which quantity is found, but not apart from all matter, as metaphysics does.

Consider, third, that this power in man is not his own making. That the intellect should see, immediately and without proof, that the whole is greater than the part, is a light given with our nature. The Psalmist says, The light of thy countenance, O Lord, is signed upon us. And because these principles are true of all created quantity, they belong to the order which Scripture ascribes to God: thou hast ordered all things in measure, and number, and weight (Wisdom 11:21). Saint Paul teaches that the invisible things of God are clearly seen, being understood by the things that are made (Romans 1:20). The steadiness of such truths, and their independence of our wishes, are among the things that are made; the mind that studies them is being trained to recognise a measure it did not set.

Consider, fourth, the place of this lesson in the liberal arts. The seven liberal arts are the three arts of language — grammar, logic, rhetoric — and the four arts of quantity — arithmetic, geometry, music and astronomy. Geometry is the art of continuous quantity. But the common notions are common to arithmetic as well as to geometry, and they are the material on which logic works, for logic teaches how conclusions follow from principles, and here you see the very principles a demonstration begins from. Thus this lesson stands at the joint of logic and the mathematical arts, and it is the plainest illustration of what a demonstrative science is: definitions, principles granted, and conclusions drawn in order.

Finally, its place in modern study and life.

The common notions are still taught in every school, though usually without their name and without their reason. Modern algebra courses list the properties of equality: the transitive property, that if a equals b and b equals c then a equals c; the addition property, that equals added to equals give equals; the subtraction property, that equals taken from equals leave equals. These are Euclid’s first, second and third common notions. Every step of solving an equation — adding the same number to both sides, subtracting the same number from both sides — is an application of common notions 2 and 3. The student who has been made to memorise these as rules without cause is being handed the conclusion of this lesson without its principle.

The fourth common notion survives in the modern treatment of congruence: figures that can be made to fit exactly are congruent, and congruent figures are equal in every measure. The fifth survives wherever a proof by contradiction ends in the absurdity that something is equal to a part of itself.

Two losses deserve mention. First, the axiomatic order of Euclid — principles first, conclusions after, nothing used before it is proved — was for two thousand years the model of orderly thought in every subject, and its abandonment in general education has left many people unable to distinguish what they have proved from what they have assumed. Second, because the modern student is rarely told that a science must rest on undemonstrated principles, he easily falls into one of two errors: he thinks that nothing is certain unless it is proved, which is impossible; or he thinks that whatever is unproved is arbitrary, which is false. The common notions correct both errors at once. They are not proved, and they are not arbitrary. They are seen.

One caution, since it is sometimes raised against the fifth notion. It is said that in the modern theory of infinite collections a whole can be matched one-to-one with a part of itself. That discussion concerns collections considered by such matching, and does not touch Euclid’s statement, which is about magnitudes — finite continuous quantities and their parts. Of the magnitudes with which geometry deals, the whole is greater than the part, and always will be.

IV. CONCLUSION

Look back at the objectives set at the beginning.

First, you were to state what a common notion is and distinguish it from a definition and a postulate. A common notion is a self-evident truth about quantity, common to more than one science and known to all men who understand its terms, accepted without demonstration and used in demonstrations. A definition fixes the meaning of a term. A postulate asks that something proper to geometry, chiefly a construction, be granted.

Second, you were to recite the five Common Notions exactly. They are given in the Memory Work below and are to be learned word for word.

Third, you were to explain the meaning of each and define the terms. Magnitude is continuous quantity; equal means neither greater nor less in quantity, and holds only between quantities of the same kind; a part is a magnitude contained in another; the whole is that which contains the part and something besides; to coincide is to be applied to another so as to fit exactly, with nothing left over on either side.

Fourth, you were to give a worked example of each. You have seen the first used in the construction of the equilateral triangle in Proposition 1; the second in the adding of equal angles in Proposition 47; the third in the taking of equal parts from equal wholes in Proposition 5; the fourth in the superposition of triangles in Proposition 4; the fifth in the reduction to absurdity in Proposition 6.

Fifth, you were to explain why first principles are not demonstrated. Because every demonstration proceeds from premises; if all premises had to be proved, the proofs would either run back without end or turn in a circle, and nothing would ever be known. Therefore there must be true, immediate, better-known premises accepted without proof, and the common notions are such.

Now study this lesson for mastery, not for acquaintance. It is short, and for that reason it is easily passed over; but every proof you will ever read in Book I rests upon these five statements, and a man who has not made them his own will follow Euclid’s demonstrations without seeing why they hold. Recite the five until you can give them exactly, in order, and without hesitation. Then take each in turn and explain it, define its terms, and produce an example of its use. Do not be content until you can teach the lesson from its causes — until you can say not only that the whole is greater than the part, but why that truth is not proved, and why the science would be impossible if it had to be. Aristotle holds that a man knows a thing when he can say why it is so. Return to this lesson until you can, and then demonstrate that mastery in the assessments set on the lesson in the Classical Liberal Arts Academy.

MEMORY WORK

A common notion is a self-evident truth concerning quantity, common to more than one science, which is accepted without demonstration and used in demonstrations.
Euclid places three kinds of principles at the head of Book I: definitions, which fix the meanings of terms; postulates, which ask that things proper to geometry be granted; and common notions, which state truths common to all quantity.
First principles are not demonstrated, because every demonstration proceeds from premises, and if all premises had to be proved, the proofs would either run back without end or turn in a circle.
Magnitude is continuous quantity, such as a line, a surface, a solid, or an angle; number is discrete quantity.
Two magnitudes are equal when neither is greater and neither is less; only magnitudes of the same kind can be equal or unequal.
Things which are equal to the same thing are also 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.
To coincide is to be applied to another so as to fit exactly, with no part of either falling outside the other; this application is called superposition.
Things which coincide are equal, but things which are equal need not coincide.

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