Number number, that abstract magnitude which, when measured, yields a multitude of unit parts, occupies a place of primary importance in the geometrical sciences. It is first defined as that which is counted by the mind, each unit being alike to the others, and the whole being a collection of such units. Hence the natural numbers arise from the successive addition of one unit to another, each addition producing a greater magnitude, the succession being infinite and without bound. The unit itself, being indivisible in the sense of the counting process, serves as the measure of all other numbers; any other magnitude is said to be a multiple of the unit when it contains a whole number of such units. From this definition follow the elementary properties which Euclid enunciates among the common notions: that equals added to equals give equals; that equals subtracted from equals give equals; that if a magnitude be greater than another, then adding to each the same magnitude preserves the inequality; and that if a magnitude be greater than another, then the greater also exceeds any part of the lesser. These notions, when applied to numbers, render the familiar arithmetical truths in a geometric language. Thus the commutativity of addition is expressed by the fact that, given two numbers a and b, the sum a + b may be represented by a line segment composed of a segment of length a followed by one of length b; the same total length is obtained when the order is reversed, for the two concatenations are congruent. The notion of multiplication is likewise rendered geometrically. To multiply a number a by a number b is to lay down b copies of a segment of length a, end to end, thereby forming a segment whose length is the product a·b. The associativity of multiplication follows from the fact that the grouping of these copies does not affect the total length, and the distributive law is seen when a segment of length a is placed alongside a segment of length b, the resulting whole being equal to the sum of the two, which in turn may be multiplied by a third number c, producing a segment whose length equals a·c plus b·c. Euclid’s treatment of numbers extends beyond the elementary operations to the study of ratios and proportion. A ratio is the relation of two magnitudes of the same kind; it is expressed by the comparison of a magnitude to another, such that a multiple of the first exceeds the second by the same amount that a multiple of the second exceeds the first. In the case of numbers, the ratio of a to b is equivalent to the fraction a⁄b, though Euclid refrains from the symbolic notation and instead employs the language of proportion: if a : b = c : d, then a·d = b·c, a fact which may be proved by constructing similar triangles, each side of which represents one of the magnitudes in question. The theory of similar figures supplies a powerful means of demonstrating the equality of ratios, for corresponding sides of similar triangles are in the same ratio, and thus the properties of numbers may be inferred from the geometry of figures. The classification of numbers into those which are prime and those which are composite occupies a distinguished place in the Elements. A prime number is defined as a number which is measured by no other number except the unit; that is, a prime admits no divisor other than one and itself. Euclid proves, by way of contradiction, that there exist infinitely many such numbers: assuming a finite list of primes, one forms the product of all these primes and adds the unit, thereby obtaining a number which is not divisible by any of the listed primes, and consequently must itself be prime or possess a prime divisor not in the original list. This argument, rendered in geometric terms, proceeds by constructing a line segment whose length equals the product of a finite set of unit lengths, then extending it by a further unit length, and observing that the resulting segment cannot be divided into an integral number of the original constituent segments, whence a new prime must arise. Numbers may also be regarded as magnitudes of the continuous kind, wherein the notion of divisibility acquires a different character. A continuous magnitude, such as a line segment, a plane surface, or a solid body, admits an infinite divisibility, for between any two points on a line there may be found another point, and so on ad infinitum. The relation between discrete and continuous magnitudes is clarified by the method of exhaustion, whereby a continuous magnitude is approximated by a sequence of inscribed polygons whose areas or lengths approach that of the whole. In this method, numbers serve as the measures of the successive approximations, and the limit of the sequence, though never attained by any finite number, is shown to coincide with the magnitude in question. Thus the area of a circle is demonstrated to be equal to that of a triangle whose base equals the circumference and whose height equals the radius, the proof relying upon the infinite succession of regular polygons inscribed within the circle, each successive polygon having twice as many sides as its predecessor, thereby rendering the perimeters and areas ever nearer to those of the circle. The concept of proportion extends further to the notion of similar figures, wherein the ratios of corresponding sides are equal, and the angles are equal. When similar triangles are constructed on a given straight line, the lengths of the sides of the triangles are in the same ratio as the numbers which measure them; consequently, the theory of similar figures provides a geometric foundation for the law of proportion among numbers. The famous theorem of the Pythagorean, which asserts that in a right‑angled triangle the square on the side opposite the right angle equals the sum of the squares on the other two sides, may be interpreted numerically as a relation among the areas of squares constructed upon the lengths of the sides. The proof, proceeding by rearrangement of the constituent squares, demonstrates that the equality of areas is a consequence of the equality of the corresponding lengths, which themselves are numbers measured by the unit length. In the study of arithmetic progressions, Euclid treats sequences of numbers each increased by a constant difference. Such a progression may be represented geometrically by a series of contiguous line segments, each longer than its predecessor by an equal amount. The sum of the terms of an arithmetic progression is shown to be equal to the product of the number of terms and the mean term, a result obtained by pairing the first and last terms, the second and penultimate terms, and so forth, each pair having the same total length. This geometric arrangement yields a visual proof of the formula for the sum, without recourse to algebraic symbols. Geometric series, wherein each term is a constant multiple of the preceding, are likewise illustrated by successive similar figures. If a line segment is divided repeatedly in a constant ratio, the resulting series of lengths forms a geometric progression. The sum of a finite geometric series is obtained by constructing a rectangle whose sides correspond to the first term and the sum of the series, and then demonstrating, through similar triangles, that the rectangle is equal in area to a larger rectangle whose dimensions are readily expressed in terms of the ratio and the number of terms. The infinite geometric series, when the common ratio lies between the unit and zero, converges to a finite magnitude, a fact established by the method of exhaustion, wherein the remainder after any finite number of terms is shown to be smaller than any given magnitude, and thus may be made arbitrarily small. The theory of numbers also embraces the notion of incommensurability, first discovered in the diagonal of a square. Two magnitudes are said to be commensurable when there exists a common measure, a unit length which measures each of them an integral number of times. The diagonal of a square, when the side is taken as the unit, is not measured by any such common unit, for the ratio of the diagonal to the side is irrational, a fact demonstrated by the impossibility of expressing the square root of two as a ratio of two integers. Euclid’s proof of this incommensurability proceeds by assuming the contrary, constructing a pair of whole numbers with no common divisor, and deriving a contradiction through the process of repeated subtraction, thereby showing that the diagonal and the side share no common measure. In the later books of the Elements, the concept of number is extended to the study of harmonic and arithmetic means. Given two magnitudes, the arithmetic mean is the magnitude which, added to each of the given magnitudes, yields equal sums; geometrically, it is represented by the midpoint of the segment joining the two magnitudes. The harmonic mean, on the other hand, is that magnitude which, when taken as a common measure, renders the reciprocals of the given magnitudes commensurable. These means are employed in the construction of proportionals and in the solution of problems concerning the division of magnitudes in given ratios. The relationship between number and geometry is further illuminated by the theory of regular polygons. The side length of a regular polygon inscribed in a circle is a number which depends upon the number of sides; as the number of sides increases without bound, the side length diminishes, and the polygon approaches the circle itself. In this limiting process, the number of sides serves as a discrete magnitude, while the circumference of the circle is a continuous magnitude. The method of exhaustion, applied to the perimeters of such polygons, yields the equality of the circumference to the product of the diameter and the constant now known as π, though Euclid refrains from assigning a numerical value to this constant, instead establishing its existence through geometric construction. Numbers also appear in the theory of proportionally similar solids, wherein volumes of similar three‑dimensional figures are in the cube of the ratio of corresponding edges. Thus, if two similar solids have corresponding edges in the ratio a : b, then the ratio of their volumes is a³ : b³. This result follows from the fact that the volume of a solid may be decomposed into a multitude of infinitesimal prisms whose heights correspond to the edges, and the multiplication of lengths in three dimensions reflects the cubic power of the ratio. The study of number in Euclid’s corpus is therefore inseparable from the study of geometry. Numbers are not abstract symbols but magnitudes measured by the unit, represented by line segments, areas, and volumes, and their properties are established by constructions, congruences, and the method of exhaustion. Through definitions, common notions, and propositions, the Elements lay down a systematic foundation for arithmetic as a branch of geometry, wherein each theorem concerning numbers is proved by means of geometric reasoning, and each geometric result may be expressed in numerical terms. This synthesis of the discrete and the continuous, of counting and measurement, constitutes the essence of the Euclidean conception of number. [role=marginalia, type=clarification, author="a.spinoza", status="adjunct", year="2026", length="49", targets="entry:number", scope="local"] Observe that the unit, though indivisible for the purpose of enumeration, is not a simple atom of being; it is a mode of the infinite attribute of quantity, whose constancy permits the notion of multiplicity, while the infinite succession of natural numbers reflects the unbounded extension of that attribute. [role=marginalia, type=heretic, author="a.weil", status="adjunct", year="2026", length="47", targets="entry:number", scope="local"] The definition treats number as a cold abstraction, yet it omits the fact that any counting presupposes a concrete, suffering world whose plurality cannot be reduced to mere units. Infinity, then, is not a boundless succession but a horizon of lack, revealing the limits of pure geometry. [role=marginalia, type=extension, author="a.dewey", status="adjunct", year="2026", length="43", targets="entry:number", scope="local"] Yet the unit’s indivisibility masks its generative ambiguity: is it a thing, a mark, or a relation? The Pythagorean monas as ontological seed gives way to the Aristotelian potential—number as predicate, not substance. This tension births algebra: number no longer counted, but composed. [role=marginalia, type=clarification, author="a.husserl", status="adjunct", year="2026", length="43", targets="entry:number", scope="local"] The unit is not merely a counted object but a noematic correlate of intentional synthesis: number arises not from things, but from the conscious act of unifying disparate intuitions into a single, identical horizon of “one-ness”—the transcendental genesis of arithmetic in pure consciousness. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:number", scope="local"] I remain unconvinced that the unit (monas) should be seen as an indivisible, qualityless entity. From where I stand, the concept of bounded rationality suggests that our perception of units is inherently complex and influenced by cognitive processes. While the unit is indeed the basis of number, its very indivisibility might be an oversimplification of how we actually process and understand numerical concepts. See Also See "Measurement" See "Number"