Quadrature of the Parabola (Τετραγωνισμὸς παραβολῆς)¶
Quadratura Parabolae — Archimedes of Syracuse (ed. T. L. Heath, 1897, pp. 233–252; J. L. Heiberg, Vol. II, pp. 294–353)
Note
Edition Track (English): Public-domain translation from heath:1897 (pp. 233–252), verified against 300 DPI scans of Heath (1897) and collated with Heiberg's critical Doric Greek edition (heiberg:1881, Vol. II, pp. 294–353). Switch tracks above for the Summary or Notes.
Prefatory Epistle to Dositheus¶
“ARCHIMEDES TO DOSITHEUS GREETING.
When I heard that Conon, who was my friend in his lifetime, was dead, but that you were acquainted with Conon and withal versed in geometry, while I grieved greatlily for the loss not only of a friend but of an admirable mathematician, I set myself the task of communicating to you, as I had intended to send to Conon, a certain geometrical theorem which had not been investigated before but has now been investigated by me, and which I first discovered by means of mechanics and then exhibited by means of geometry.”
Archimedes recounts to Dositheus how earlier geometers attempted in vain to square the circle and hyperbolic or elliptic segments, whereas no one had hitherto attempted to square the segment bounded by a straight line and a section of a right-angled cone (parabolic segment). He announces its exact quadrature—that any parabolic segment is four-thirds of the triangle having the same base and equal height—and explicitly states the continuity postulate employed in the proof (Archimedes' Lemma, cf. hilbert:1899):
Archimedes notes that earlier geometers (Eudoxus of Cnidus) used this very lemma to prove that circles are to one another in the duplicate ratio of their diameters (Eucl. XII. 2), spheres in the triplicate ratio of their diameters (Eucl. XII. 18), every pyramid one-third of the prism on the same base and equal altitude (Eucl. XII. 7), and every cone one-third of the cylinder (Eucl. XII. 10).
Part I — Mechanical Exhaustion via the Lever (Propositions 1–17)¶
Propositions 1–5: Conic Symptomata of the Parabola¶
Drawing on the pre-Apollonian Elements of Conics (Euclid and Aristaeus), Archimedes recalls for a parabola with chord QQ bisected at V by diameter PV and tangent PT at vertex P:
Prop. 1–2: bisects all chords parallel to QQ, and if tangent QT meets diameter PV produced at T, then .
Prop. 3: Ordinate-squared property .
Props. 4–5: For any chord Qq, diameter PV through P, and parallel lines drawn to the diameter meeting the tangent qT at E, the parabola at R, and the chord Qq at O (while qO meets PV at V), the intercept ratio satisfies:
\[QO : Oq = ER : RO.\]
Propositions 6–15: Balancing Triangles and Trapezia on a Beam¶
Suspend a right-angled triangle BDG beneath a horizontal balance beam AOB with fulcrum O and equal opposite arm :
Props. 6–7: Because the centre of gravity of triangleBDG lies horizontally at from O (Equil. I. 14), an area P hung at A balances triangleBDG in its actual position when .
Props. 8–13: Establishes upper and lower lever inequalities for every vertical trapezium strip cut from triangleBDG by equally spaced parallels to the diameter.
Props. 14–15: Partitioning the base Qq of the parabolic segment into equal parts and drawing parallels to the diameter inscribes and circumscribes finite families of rectilinear trapezia inside triangleQqT (whose area is ). Summing the lever inequalities across all strips compresses the balancing weight M at A between circumscribed and inscribed trapezium weights whose difference equals a single triangle of base , which can be made smaller than any assigned area (Archimedes' Lemma).
Propositions 16 & 17: First Quadrature Theorem¶
Proposition 16: By double indirect exhaustion on the balance beam, the parabolic segment equals one-third of the enveloping tangent triangle triangleQqT.
Proposition 17: Since (where P is the vertex of the segment where the tangent is parallel to Qq), it follows immediately that:
The area of any segment of a parabola is four-thirds of the triangle which has the same base and equal height.
Part II — Pure Geometric Exhaustion (Propositions 18–24)¶
To demonstrate the theorem purely geometrically without mechanical postulates, Archimedes defines the base AC, vertex B (point of maximum perpendicular height, where the tangent is parallel to AC), and height of any segment:
Propositions 18–19: If diameters through the midpoints F, G of AD (half-chord) meet the parabola at secondary vertices D, E, then (actually measured from the chord, so the sub-segment height is of the main height measured along the diameter!).
Proposition 20: If ABC be a parabolic segment with vertex B, then triangleABC is greater than half the area of the segment ABC (since triangleABC is half of the circumscribed parallelogram formed by the tangent at B and diameters through A, C). Repeated inscription therefore exhausts the segment by Eucl. X. 1!
Proposition 21: If D, E be the vertices of the two remaining segments on AB and BC, then:
\[\triangle ABC = 8 \cdot \triangle ADB = 8 \cdot \triangle BEC, \qquad \text{whence} \qquad \triangle ADB + \triangle BEC = \frac{1}{4} \triangle ABC.\]Proposition 22: For any finite sequence of areas ABCDdots formed at successive stages of this bisection, their sum is strictly less than the area of the parabolic segment.
Proposition 23 (Exact Finite Geometric Series Identity):
Given a series of areas ABCDdotsZ, of which A is the greatest, and each is equal to four times the next in order, then:
\[A + B + C + D + \dots + Z + \frac{1}{3} Z = \frac{4}{3} A.\]Proof: Add , , …, . Since , , …, , summing all terms gives . Subtracting from both sides leaves !
Proposition 24 (Geometric Quadrature of the Parabola):
Every segment bounded by a parabola and a chord AC is equal to four-thirds of the triangle ABC which has the same base as the segment and equal height.Let .
Suppose the segment greater than K: Inscribe triangles recursively until the remaining parabolic segments leave a sum less than (Prop. 20, Eucl. X. 1). Then the inscribed polygon , contradicting Prop. 23 ()!
Suppose the segment less than K: Let , , …, continuing until the last term Z is less than . By Prop. 23, , whence , contradicting Prop. 22!
Therefore the parabolic segment is neither greater nor less than ; it is strictly equal to .