CLAA – Eucild's Elements - Book I, Definitions

QRV-321 Lesson 001 – Euclid Elements, Book I, Definitions

I. INTRODUCTION

This lesson begins the study of geometry, and it begins where every science must begin: with definitions. Geometry is the science of continuous magnitude — that is, of extension, of what has size and shape. Arithmetic studies multitude, which is number, and number is discrete: it is made of units that are separate from one another. Geometry studies magnitude, which is continuous: its parts hold together and are not separated by anything. Lengths, surfaces, and solids are continuous magnitudes, and geometry is the science of them.

Before anything can be proved about magnitude, the terms used in the proofs must be fixed. That is what these twenty-three definitions do. They are not propositions to be proved. They are principles — starting points — which state plainly what the words of the science signify. Nothing in geometry can be demonstrated until it is settled what a line is, what a right angle is, what a circle is. So this lesson stands first, before the postulates, before the common notions, and before the first proposition.

The text before you is the opening of the Elements of Euclid. Euclid was a Greek mathematician who taught at Alexandria in Egypt around 300 B.C., in the generation after Aristotle, under the first of the Ptolemies. He did not invent most of what is in the Elements. The doctrine came down to him from Thales, Pythagoras and the Pythagoreans, Hippocrates of Chios, Eudoxus of Cnidus, Theaetetus, and others. What Euclid did was to gather this scattered inheritance and set it in order: to reduce it to a small number of definitions, postulates and common notions, and then to demonstrate everything else from them, each proposition resting only on what had already been established. This is why his book is called the Elements — it sets out the elements, the first parts, out of which the whole science is built.

The Elements has been the single most used textbook in the history of the world. It was studied continuously in the Greek schools; Theon of Alexandria in the fourth century after Christ produced the recension from which most of our manuscripts descend, and one great manuscript in the Vatican preserves an older text than Theon’s. Proclus wrote a commentary on Book I in the fifth century which preserves much of the earlier history of the science. The Arabs translated it; from the Arabic, Adelard of Bath rendered it into Latin about the year 1120, and Campanus of Novara made the version that was printed first, at Venice by Erhard Ratdolt in 1482. Later Latin editions by Commandino and by the Jesuit Christopher Clavius carried it through the Catholic schools of the sixteenth and seventeenth centuries. The English wording given in this lesson follows the standard translation of Sir Thomas Heath (1908), which is made directly from the Greek. The book has been kept for two reasons: because its doctrine is true, and because its order is the clearest example we possess of a science demonstrated from its principles. Aristotle’s account of what a science is, given in the Posterior Analytics, is best seen at work in Euclid.

One caution belongs at the start. Euclid’s definitions do not assert that these things exist, and they do not describe things floating in a separate world of their own. Following Aristotle, and the Scholastic doctors after him, we hold that the geometer knows lines and figures by abstraction from sensible things. You see bodies with your eyes. You touch them with your hands. The intellect, considering such a body, sets aside its color, its weight, its material, its motion, and considers only its extension and shape. What remains under that consideration is the object of geometry. A point, a line, a surface are therefore truly found in bodies, but the geometer considers them apart from the sensible qualities in which they are found. He does not consider them apart from all matter whatsoever, for a triangle requires extension; but he considers them apart from this or that particular sensible matter. This is what the Scholastics call the second degree of abstraction: natural science considers changeable bodies as changeable; mathematics considers quantity abstracted from sensible qualities; metaphysics considers being as being, apart from matter altogether.

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

State the twenty-three definitions of Euclid, Book I, in their proper form.
Explain what a definition is in a science, and why definitions are principles and not conclusions.
Explain the three magnitudes — line, surface, solid — by the dimensions each possesses, and name the extremities of each.
Define angle, right angle, perpendicular, obtuse angle and acute angle, and explain why Euclid defines a right angle without using degrees.
Define boundary, figure, circle, centre, diameter and semicircle, and show how the later definitions depend on the earlier ones.
Classify rectilineal figures by the number of their sides, triangles by their sides and by their angles, and quadrilaterals into the five kinds Euclid names.
Define parallel straight lines and explain the three conditions in that definition.
Distinguish Euclid’s terms from the modern terms that have replaced some of them.

II. EXPOSITION

First, what is a definition?

A definition is a statement that makes known what a thing is, or what a name signifies. In the ordinary and best form, taught by Aristotle, a definition is made of two parts: the genus, which is the wider class to which the thing belongs, and the difference, which marks off this thing from everything else in that class. Thus “a square is a quadrilateral figure which is both equilateral and right-angled.” The genus is “quadrilateral figure”; the difference is “equilateral and right-angled.”

In a demonstrative science, definitions are principles. A principle is that from which something proceeds and which is not itself derived from anything prior in that science. You cannot prove a definition, because a proof must use terms, and the terms must already be defined. Nor can definitions go on forever, for then nothing would ever be known. So a science must begin with terms whose meaning is given, and Euclid gives them here.

You will notice that the first definitions are not of the genus-and-difference form. A point cannot be put under a wider class of magnitude, because it is not a magnitude at all. So Euclid defines it by removing something: it has no part. Such definitions by negation or by removal are proper where the thing defined is first and simplest.

Notice also the order. Euclid begins with what is simplest and least, the point, and builds upward: point, line, surface. He then defines angle from lines, figure from boundary, circle from plane figure and equal straight lines, semicircle from diameter and circumference, kinds of triangle from trilateral figure, and so on. Every later definition uses only terms already defined. This is the order of a science.

Now take the definitions in Euclid’s own order.

A point is that which has no part.

The Greek word is sēmeion, a sign or mark. “Part” here means a portion of extension. To have no part is to be indivisible: you cannot cut a point in two, because there is nothing in it to cut. It has no length, no breadth, no thickness. It therefore has no magnitude at all.

Then what is a point? It is the extremity, or terminus, or limit, of magnitude. It has position, but no size. The corner of a room is a point; the tip of a needle approaches one; the meeting of two edges of a table is a point. In each case what you actually see or touch is a little bit of matter, because your eyes cannot see what has no size. But the intellect grasps the limit itself, and that limit is what is meant.

Do not think a line is made by heaping up points, as a heap of sand is made of grains. A point has no length; no number of things having no length can amount to length. The point is the end of a line, not a piece of it.

A line is breadthless length.

“Length” means extension in one direction. “Breadthless” means without any width. So a line has one dimension only. A dimension is a direction of extension. A line has length and nothing else.

Again, the line you draw with a pen has breadth, because ink has breadth. The geometer does not consider the ink. He considers the length only, abstracting from the breadth. The edge where a wall meets a floor is a line; the crease where a sheet of paper is folded is a line. Considered exactly, these have length and no breadth.

Note that “line” in Euclid does not mean “straight line.” A line may be straight or curved. The circumference of a circle is a line. This is why Euclid must give a separate definition of the straight line in number 4.

The extremities of a line are points.

An extremity is an end, a bounding limit. If a line is bounded — if it has ends, as a segment does — then those ends are points. They are not short lines. A limit is of a lower kind than the thing it limits: what bounds length is not itself length.

Some lines have no extremities: the circumference of a circle returns into itself and has no end point. So the definition tells us what the extremities of a line are when it has them.

A straight line is a line which lies evenly with the points on itself.

This is Euclid’s hardest early definition, and it must be read carefully. “Lies evenly” translates the Greek ex isou keitai — it lies equally, or uniformly. The meaning is that a straight line does not deviate at any of its points. Take any two points on it: the part of the line between them holds the same direction throughout; no point of it stands aside from the rest. Every part of a straight line is placed with respect to its points exactly as every other part is; the line has one and the same direction at every point of itself.

Restated plainly: a straight line is a line which does not bend at any point; it keeps one direction from end to end, so that all its points lie evenly, none swerving out of line with the others.

Other definitions of the straight line have been given. Plato is reported to have said it is the line whose middle points cover the extremes when the line is viewed end-on. Archimedes assumed that the straight line is the shortest of all lines having the same extremities. Euclid does not use either. He defines straightness by the evenness of the line’s own points, and he does not need the notion of shortest, which he proves later as a theorem instead (Book I, Proposition 20).

A surface is that which has length and breadth only.

A surface has two dimensions: length and breadth. “Only” excludes the third, thickness or depth. The face of a wall is a surface; the top of still water is a surface; the outside of a ball is a surface. What you see has some thickness of matter under it, but the surface itself is the boundary, and boundaries have no thickness.

Here we can see the whole ordered series of magnitudes:

A point has no dimension. A line has one dimension: length. A surface has two dimensions: length and breadth. A solid has three dimensions: length, breadth, and depth. (Euclid defines the solid later, in Book XI, because Books I through VI treat of plane figures.)

Each is bounded by the one before it.

The extremities of a surface are lines.

Just as a bounded line ends in points, a bounded surface is bounded by lines. The four edges of a rectangular sheet are the extremities of its surface. The circumference is the extremity of the surface of a circle. Again the limit is of a lower dimension than what it limits: what bounds a two-dimensional surface is one-dimensional.

Some surfaces have no extremities, as the whole outer surface of a sphere, which closes upon itself.

A plane surface is a surface which lies evenly with the straight lines on itself.

This corresponds exactly to definition 4, raised one dimension. A plane, or flat, surface is one which nowhere bulges or dips. Euclid expresses this by the straight lines lying in it: take any straight line lying in the surface, and the surface lies evenly with it, that is, it does not depart from it in any direction; it holds the same disposition all along. A useful consequence, which is what the definition amounts to in practice, is this: if two points of a straight line lie in a plane, the whole of that straight line lies in that plane.

Restated plainly: a plane surface is a flat surface, one that keeps the same evenness throughout, so that straight lines drawn in it lie flat in it along their whole length.

The surface of a wall or a table is a plane surface. The surface of a ball or of a cylinder is a surface but not a plane surface, for a straight line touching it does not lie in it.

A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

An angle is not the two lines. It is not the point where they meet. It is the inclination — the leaning of one toward the other, the amount of opening between them at the point where they meet.

The definition has four parts, and each is needed:

“the inclination to one another of two lines” — an angle requires two lines, which are said to contain the angle. “in a plane” — hence the name plane angle. Both lines lie in one flat surface. Euclid adds “plane” because there are other kinds of angle, such as the solid angle at the corner of a cube, treated in Book XI. “which meet one another” — lines that never touch contain no angle. The point where they meet is called the vertex of the angle. “and do not lie in a straight line” — if the two lines meet and together form one straight line, there is no inclination of one to the other; they lean no way at all. So Euclid excludes this case from the notion of angle.

Notice that Euclid says “two lines,” not “two straight lines.” Curved lines meeting in a plane also make an angle; the angle between a circle and its tangent, or between two arcs, is an angle in Euclid’s sense. That is why the next definition is needed.

And when the lines containing the angle are straight, the angle is called rectilineal.

Rectilineal means straight-lined, from the Latin rectus, straight, and linea, line. A rectilineal angle is an angle contained by two straight lines. Nearly all the angles in Book I are rectilineal.

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Read this slowly. Take a straight line. Set another straight line upon it, meeting it, so that two angles are formed, one on each side of the standing line. These two are called adjacent angles, because they lie next to each other, share the vertex, and share the standing line as a common side. Now, if those two adjacent angles are equal to one another, then each of them is called a right angle; and the standing line is called a perpendicular to the line it stands on.

Notice three things.

First, Euclid does not say a right angle is an angle of ninety degrees. He defines it by equality alone. Degrees are a measure taken over from Babylonian astronomy, in which the circle was divided into three hundred and sixty parts; the division is useful, but it is a convention of measurement, not the nature of the thing. Euclid’s definition needs no unit and no number. It says: the right angle is the angle that equals its own adjacent angle. This is a purely geometrical definition of continuous magnitude, and it does not depend on any chosen standard.

Second, the definition gives you two things at once: the right angle, and the perpendicular. A perpendicular is not a kind of line by itself; it is a line related to another line. It is always perpendicular to something.

Third, the definition alone does not prove that all right angles anywhere are equal to each other. Euclid does not leave that to chance: he lays it down separately as the fourth postulate. That is a mark of his care. A definition tells you what a word means; it cannot by itself establish a truth about all cases.

An obtuse angle is an angle greater than a right angle.

Obtuse means blunt. Its genus is angle; its difference is being greater than a right angle. Since the right angle is now fixed, greater and less can be spoken of.

An acute angle is an angle less than a right angle.

Acute means sharp. So rectilineal angles fall into three kinds by comparison with the right angle: acute, right, and obtuse. Every angle is one of these three, and no angle is two of them.

A boundary is that which is an extremity of anything.

Boundary translates the Greek horos, a limit or term. A boundary is a limit: that at which a thing stops. Definitions 3 and 6 already gave instances. The points at the ends of a segment are its boundaries; the lines around a surface are its boundaries. Definition 13 gives the general name, so that it may be used in the next definition.

A figure is that which is contained by any boundary or boundaries.

A figure, Greek schēma, is an enclosed magnitude: it is what is shut in by one boundary or by several. “Contained by” means enclosed on all sides, so that the figure is complete and closed.

Notice “any boundary or boundaries.” Some figures are contained by one boundary only — the circle is contained by one line. Others are contained by several — a triangle by three straight lines, a quadrilateral by four.

Notice also that a figure must be closed. Two straight lines meeting at a point do not contain a figure; there is a gap. Three straight lines can enclose a plane figure, but two straight lines cannot; that is why the triangle is the least of the rectilineal figures.

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;

Take this apart.

“a plane figure” — the genus. It is enclosed, and it lies in a plane. “contained by one line” — the boundary is a single line, which returns into itself. This one line is called the circumference of the circle. “such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another” — the difference. There is some one point inside the figure with this property: every straight line drawn from that point to the boundary is equal to every other such line. Each of those equal straight lines is called a radius.

Restated plainly: a circle is a closed flat figure whose single boundary is everywhere the same distance from one interior point.

Two remarks. First, for Euclid the circle is the whole figure, boundary and enclosed surface together; the boundary line alone is the circumference. Modern usage often calls the boundary line itself the circle, and it is worth knowing the difference so that Euclid’s propositions are read rightly. Second, “from one point among those lying within the figure” is exact language: not from any interior point whatsoever, but from one particular interior point. If you take any other interior point, the straight lines from it to the circumference are not all equal.

And the point is called the centre of the circle.

The point spoken of in definition 15 receives its name here: the centre. So the centre of a circle is that interior point from which all straight lines drawn to the circumference are equal.

A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

A diameter must satisfy three conditions: it is a straight line; it passes through the centre; and both its ends fall on the circumference, so that it is terminated, that is, bounded, by the circumference at each end. “In both directions” means at each of its two ends.

A circle has infinitely many diameters, since a straight line can be drawn through the centre in any direction; hence Euclid says “any straight line.” Because the diameter is made of two radii laid in a straight line, every diameter is double the radius, and all diameters of the same circle are equal.

“And such a straight line also bisects the circle.” To bisect is to cut into two equal parts. Euclid states here that the diameter divides the circle into two equal parts. Strictly this is a property, not part of the meaning of the word, and Proclus records that Thales was the first to demonstrate it. Euclid places it here because he needs it at once for the next definition.

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

Semi means half. A semicircle is a figure — so it is closed — and it is contained by two boundaries: one straight, namely the diameter, and one curved, namely the arc, that is, the part of the circumference which the diameter cuts off. The arc together with the diameter encloses the semicircle.

“And the centre of the semicircle is the same as that of the circle.” The middle point of the diameter, which is the centre of the whole circle, is also called the centre of the semicircle. This is stated so that the word “centre” may be used of a semicircle without ambiguity.

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Here figures are divided by the kind and number of their boundaries. Rectilineal figures are those whose boundaries are all straight lines. They are then divided by number:

trilateral — three sides. Latin tres, three, and latus, side. quadrilateral — four sides. Latin quattuor, four. multilateral — more than four sides. Latin multus, many.

“Side” means one of the straight lines containing the figure. Euclid names these classes by their sides, not by their angles; the word “triangle,” which names the figure from its three angles, he uses freely as well, and the two names apply to the same figures, since a figure contained by three straight lines has three angles.

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

This divides triangles by their sides.

Equilateral: from Latin aequus, equal, and latus, side. All three sides equal. Isosceles: from Greek isos, equal, and skelos, leg. Two sides equal. Scalene: from Greek skalēnos, uneven. No two sides equal.

Attend to the word “alone” in the definition of the isosceles triangle: “two of its sides alone equal.” For Euclid, the isosceles triangle has exactly two equal sides, and the equilateral triangle is therefore not counted as isosceles. Many modern books define the isosceles triangle as having at least two equal sides, so that the equilateral is a special isosceles. Know Euclid’s usage when reading Euclid.

This division is exhaustive: any triangle has either three sides equal, or two alone equal, or no two equal. There is no fourth case.

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

This divides triangles again, now by their angles. The same triangle falls under one head of definition 20 and one head of definition 21; for instance, a triangle may be both isosceles and right-angled.

Note the difference in the wording. For the first two Euclid says “has a right angle” and “has an obtuse angle” — one is enough, and one is all that is possible, since a triangle cannot have two right angles or two obtuse angles, nor one of each. But for the third he says “has its three angles acute,” because having one acute angle proves nothing: every triangle has at least two acute angles. Only when all three are acute is the triangle called acute-angled. Euclid’s care in wording is exact and deliberate.

The side opposite the right angle in a right-angled triangle is called the hypotenuse, from the Greek for “stretched under.”

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Here “equilateral” means having all four sides equal, and “right-angled,” said of a quadrilateral, means having all four angles right. With those two marks, taken and denied, Euclid forms his classes:

A square: all four sides equal, and all four angles right. An oblong: all four angles right, but the sides not all equal. This is what is now commonly called a rectangle that is not a square. A rhombus: all four sides equal, but the angles not right. It is sometimes described as a slanted square. A rhomboid: not equilateral and not right-angled, but with opposite sides equal to one another and opposite angles equal to one another. This is what is now commonly called a parallelogram that is neither a rhombus nor a rectangle. Trapezia: the name given to every other quadrilateral, that is, to all those lacking these properties. The singular is trapezium.

Two remarks. First, Euclid’s classes are exclusive: for him a square is not called an oblong, and an oblong is not called a rhomboid. Modern textbooks usually make the classes inclusive, so that a square is counted as a rectangle, a rhombus, and a parallelogram. Euclid’s names mark off distinct kinds. Read him by his own definitions.

Second, “let quadrilaterals other than these be called trapezia” is an act of naming, not a description of a nature. Euclid is providing a word so that the remaining cases can be spoken of.

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Parallel is from the Greek parallēlos, alongside one another. To produce a line means to extend it further in its own direction. “Indefinitely” means as far as you please, without stopping.

Three conditions must all be met:

First, both must be straight lines. Second, they must be in the same plane. Two straight lines in different planes may fail to meet without being parallel; such lines are now called skew lines. Euclid’s condition rules them out. Third, however far they are produced, in either direction, they never meet.

Restated plainly: two straight lines in one plane are parallel when, extended forever both ways, they never touch each other.

Notice that this definition is negative: it tells you what parallel lines do not do. It gives no means of finding out whether two given lines are parallel, because no man can follow lines out indefinitely to see. For this reason Euclid must lay down, as his fifth postulate, a positive condition under which two straight lines will meet. From that postulate the whole doctrine of parallels in Book I is drawn. Here, only the meaning of the word is fixed.

III. REFLECTION

These twenty-three definitions perfect the intellect in its first and simplest act. The Scholastic doctors distinguish three acts of the intellect: simple apprehension, by which we grasp what a thing is and express it in a term or a definition; judgment, by which we join or divide terms and express the result in a proposition; and reasoning, by which we pass from propositions already known to a conclusion, and express this in an argument. Definitions belong to the first act. They are the beginning of all knowledge, for what is not first grasped cannot be judged, and what is not judged cannot be reasoned about. Euclid’s whole book rests upon this page.

Geometry itself is one of the four mathematical arts of the quadrivium: arithmetic, which treats of number in itself; music, which treats of number in relation; geometry, which treats of magnitude at rest; and astronomy, which treats of magnitude in motion. The quadrivium follows the trivium — grammar, logic and rhetoric — because you must be able to read a text, to define a term, and to follow an argument before you can follow a demonstration. And the quadrivium precedes natural philosophy and metaphysics, because the mind trained on quantity, where demonstration is clearest, is then able to reason about matters where it is harder.

In the order of the speculative sciences, mathematics stands in the middle. Natural science considers bodies as changeable, and cannot leave sensible matter behind. Metaphysics considers being as being, and leaves matter behind altogether. Mathematics stands between: it considers quantity abstracted from sensible qualities, but not from all extension. This is why geometry is the fittest training for a beginner in demonstrative reasoning. Its objects are drawn from the senses, so they are easy to grasp; yet they are abstract, so that what is proved of them is proved universally and with certainty. When you prove something of a triangle, you have proved it of every triangle that ever was or will be.

Hold firmly to the Aristotelian account of how you know these things. You do not remember points and lines from some previous acquaintance with a separate world of mathematical objects, as the Platonists taught. You see and touch bodies; the intellect abstracts from them the quantity and shape they truly have; and it is of that abstracted quantity that geometry speaks. There is no point without position, no line except as the boundary of a surface or the path between limits, no triangle except as the shape of something extended. Nothing in these definitions asks you to leave the sensible world. They ask you to consider one aspect of the sensible world exactly.

This study also serves the knowledge of God. Scripture says of God: “thou hast ordered all things in measure, and number, and weight” (Wisdom 11:21). And Saint Paul teaches that “the invisible things of him, from the creation of the world, are clearly seen, being understood by the things that are made” (Romans 1:20). When you learn that a circle is one line everywhere equally distant from one point, and that this holds of every circle without exception, you learn something of the order that God has placed in creation. The regularity is not of your making. You did not decide that the three angles of a triangle should be what they are. You found it, and you found it necessary. The steadiness and exactness of such truths is a real sign of the wisdom of the Creator, who made all things in measure and number and weight. This is a sober and traditional use of geometry, and it was one of the reasons the Fathers and the Scholastics prized the study.

As for modern use, it is wide and easy to see. Every art that works with shape and measure depends on these definitions: surveying, carpentry, masonry, architecture, drafting, machining, engineering, navigation, optics, cartography, and the geometry used in computer drawing and design. The words themselves survive in daily speech: point, line, plane, angle, right angle, perpendicular, radius, diameter, circumference, triangle, square, rhombus, parallel. Set squares, plumb lines, and the carpenter’s practice of testing a corner all rest on definition 10.

Two losses should be named plainly. First, most modern schooling teaches geometrical facts and formulas without demonstration. A student learns that the area of a circle is found by a formula, but never learns why, and never learns to prove anything at all. He gains information and loses the science. Second, exact definition has largely disappeared from schooling in every subject. A student is asked to give examples, or impressions, or applications, but rarely to say precisely what a thing is. Euclid’s first page is a remedy for both losses. It shows that a science begins by saying exactly what its terms mean, and that everything afterward must be proved from what has already been granted. That habit is worth as much outside geometry as within it.

IV. CONCLUSION

Consider now the objectives set out at the beginning.

First, you were to state the twenty-three definitions in their proper form. Each has been given, explained word by word, and set in its place in Euclid’s order.

Second, you were to explain what a definition is and why definitions are principles. You have learned that a definition makes known what a thing is or what a name signifies, that in its full form it consists of genus and difference, and that definitions cannot be proved, since all proof uses terms whose meaning must first be settled.

Third, you were to explain the three magnitudes by their dimensions and name their extremities. A line has length only, and its extremities are points; a surface has length and breadth only, and its extremities are lines; a solid has length, breadth and depth. The point, having no part, is not a magnitude but the limit of magnitude.

Fourth, you were to define angle, right angle, perpendicular, obtuse and acute angles, and to explain why Euclid defines the right angle without degrees. An angle is the inclination of two lines meeting in a plane and not lying in a straight line; a right angle is one of two equal adjacent angles made by one straight line set up on another, and the standing line is a perpendicular; obtuse is greater and acute is less than a right angle. Degrees are a convention of measurement taken from Babylonian astronomy; Euclid’s definition rests on equality alone and needs no unit.

Fifth, you were to define boundary, figure, circle, centre, diameter and semicircle, and to see how they depend on one another. A boundary is an extremity; a figure is what is contained by boundaries; a circle is a plane figure contained by one line, with one interior point from which all straight lines to that line are equal; that point is the centre; a diameter passes through the centre and is terminated both ways by the circumference, and bisects the circle; a semicircle is contained by a diameter and the arc it cuts off.

Sixth, you were to classify rectilineal figures. They are trilateral, quadrilateral, or multilateral by number of sides; triangles are equilateral, isosceles or scalene by their sides, and right-angled, obtuse-angled or acute-angled by their angles; quadrilaterals are square, oblong, rhombus, rhomboid, or trapezia.

Seventh, you were to define parallel straight lines and explain the three conditions. They must be straight, they must lie in the same plane, and they must never meet however far they are produced in either direction.

Eighth, you were to distinguish Euclid’s terms from modern ones. Euclid’s isosceles has exactly two equal sides; his circle is the whole figure, not the boundary line only; his oblong is a non-square rectangle, and his rhomboid a parallelogram that is neither equilateral nor right-angled; and his classes exclude one another where modern classes usually include one another.

Now study this lesson for mastery, not for acquaintance. Acquaintance means you have read the definitions and recognize them. Mastery means you can state each definition exactly from memory, explain every word in it, say what it depends upon and what depends upon it, and give a literal example of it. Nothing that follows in geometry can be understood without this. A proposition proved about an isosceles triangle will mean nothing to a reader who cannot say what an isosceles triangle is.

Return to the lesson until you can teach its content from its causes — that is, until you can say not only what each definition states, but why Euclid states it just so, why it stands where it stands, and what would go wrong if it were dropped or loosely worded. Aristotle teaches that we know a thing when we know why it is so. Then develop and demonstrate that mastery through the assessments set on this lesson in the Classical Liberal Arts Academy.

MEMORY WORK

A definition is a statement that makes known what a thing is or what a name signifies, and in its full form it consists of the genus and the difference.
Definitions are principles of a science; they are not proved, because every proof must use terms whose meaning is already given.
A point is that which has no part.
A line is breadthless length, and the extremities of a line are points.
A straight line is a line which lies evenly with the points on itself.
A surface is that which has length and breadth only, and the extremities of a surface are lines.
A plane surface is a surface which lies evenly with the straight lines on itself.
A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line; and when the lines containing the angle are straight, the angle is called rectilineal.
When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
An obtuse angle is an angle greater than a right angle, and an acute angle is an angle less than a right angle.
A boundary is that which is an extremity of anything, and a figure is that which is contained by any boundary or boundaries.
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 that point is called the centre of the circle.
A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
A semicircle is the figure contained by the diameter and the circumference cut off by it, and the centre of the semicircle is the same as that of the circle.
Rectilineal figures are those which are contained by straight lines; trilateral figures are contained by three, quadrilateral by four, and multilateral by more than four straight lines.
Of trilateral figures, an equilateral triangle has its three sides equal, an isosceles triangle has two of its sides alone equal, and a scalene triangle has its three sides unequal.
Of trilateral figures, a right-angled triangle has a right angle, an obtuse-angled triangle has an obtuse angle, and an acute-angled triangle has its three angles acute.
Of quadrilateral figures, a square is both equilateral and right-angled; an oblong is right-angled but not equilateral; a rhombus is equilateral but not right-angled; a rhomboid has its opposite sides and angles equal to one another but is neither equilateral nor right-angled; and quadrilaterals other than these are called trapezia.
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

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