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Geometry · magnitude at rest

Cut a cone, find three curves.

Geometry is the art of magnitude at rest. This lab follows a single work, the Conics of Apollonius of Perga, written in the late third century BC in eight books, seven of which survive. Six figures let you cut a cone with a plane, lay Apollonius' areas along a line until they fit, fall short or overshoot, move a point round the two foci, and change one number, the eccentricity, to pass from a circle to a hyperbola. Each figure is a model, and each says what it leaves out.

Lengths are in abstract units and areas in square units (sq units in the readouts). All angles are in degrees. Numbers a figure computes are marked 'computed here' in the sources.

Part of Geometry in the curriculum, the art of magnitude at rest.

I · The cone

One cone gives all three curves.

Apollonius of Perga worked in the late third century BC. Heath (1896) calls the record of his life 'the most meagre information': he was born at Perga in Pamphylia in the reign of Ptolemy Euergetes (247 to 222 BC), studied at Alexandria under the successors of Euclid and flourished under Ptolemy Philopator. In 1921 Heath revised this: Apollonius flourished under Euergetes and was 'probably born about 262 B.C.' The sources give no secure birth year. His contemporaries, Geminus reports, called him 'the great geometer' for the Conics. Of its eight books, I to IV survive in Greek and V to VII only in Arabic; Book VIII is lost. Book I opens with a cone made by motion. A straight line through a fixed point moves round a circle that is not in the same plane as the point, and traces a double cone, two cones meeting apex to apex. The cone is oblique in general, and right only when its axis is perpendicular to the base. From one such cone, by changing the cutting plane, Apollonius produces all three sections (I.11 to 13). In I.14 a plane that cuts both halves of the double cone, without passing through the apex, gives a hyperbola in each, and he calls the pair 'opposite' sections; in his usage 'hyperbola' means one branch. His preface says Book I contains 'the modes of producing the three sections and the opposite branches [of the hyperbola] ... and their fundamental properties worked out more fully and generally than in the writings of other authors'.

Earlier geometers worked differently. Eutocius, quoting Geminus, says the ancients knew only right cones. They sorted them by the angle at the apex, less than, equal to or greater than a right angle, and cut each kind by a plane perpendicular to one of its generating lines, the straight lines through the apex that make up the cone. Each kind gave one curve, named after its cone: the 'section of an acute-angled cone' (the ellipse), 'of a right-angled cone' (the parabola) and 'of an obtuse-angled cone' (the hyperbola). Menaechmus, a pupil of Eudoxus in the middle of the fourth century BC, is credited with discovering the curves, on the strength of Eratosthenes' advice not 'to cut the cone in the triads of Menaechmus'. How he came to cut a cone at all, Heath writes, 'we have no information whatever'. Eutocius credits Apollonius with showing that 'in every cone, whether right or scalene, all the sections are found'. Heath disputes the priority. Euclid knew that a cone, which Heath presumes right, or a cylinder, cut by a plane not parallel to the base, gives an ellipse when the section lies wholly between the apex and the base. Archimedes states that every section of a cone, even an oblique one, that meets all its generating lines is a circle or an ellipse, and Heath infers that whoever found this used a method that would equally give the parabola and the hyperbola. So, Heath concludes, the claim 'rests on a misapprehension'. Yet 'in the generality of his treatment of the subject from the very beginning, Apollonius was making an entirely new departure'.

The figure draws a right double cone, the special case, and turns the cutting plane about a line that touches the cone at one point, so that point is always a vertex of the section. The upper view shows the cone, the plane and the section in three dimensions, with hidden parts dashed. The strip below, 'The section, seen face-on', shows the section's true shape in its own plane and labels the two branches of a hyperbola 'opposite sections'. Three sliders set the Tilt of the cutting plane (0 to 90 degrees), the Half-angle of the cone (15 to 60 degrees) and Turn the view; dragging the cone sideways also turns it. The buttons Circle, Ellipse, Parabola and Hyperbola set the tilt for each section, and Sweep the plane runs the tilt from 0 to 90 degrees and back. The readouts give the Section, the Eccentricity (the fixed ratio of section VI), which for a right cone is sin(tilt) ÷ cos(half-angle), and the Plane against the cone's side, which is the tilt minus (90 degrees − half-angle). A level plane cuts a circle. A plane shallower than the cone's side closes the curve into an ellipse, a plane parallel to the side gives a parabola, and a steeper plane also cuts the other half of the cone and gives a hyperbola with two branches. For what geometry agrees to grant before it proves anything, see the Academy on Euclid's postulates.

Try this

  • Leave the Half-angle of the cone at 30° and press Circle, Ellipse, Parabola and Hyperbola in turn. Eccentricity reads 0.000, 0.577, 1.000 and 1.150, and Plane against the cone's side reads shallower by 60.0°, shallower by 30.0°, parallel and steeper by 25.0°.
  • Start from the defaults, a tilt of 40.0° on a half-angle of 30°: the section is an ellipse of eccentricity 0.742, shallower by 20.0°. Raise the tilt to 60.0° and it becomes a parabola. At 60.5° it is already a hyperbola, eccentricity 1.005: only the exact tilt gives a parabola. Its second branch, the opposite section, lies beyond the part of the cone drawn, as the note in the strip says; raise the tilt to 71.0° and its tip comes into view.
  • Set the half-angle to 45° and press Parabola: the tilt moves to 45.0°, because for a parabola the tilt and the half-angle add up to 90 degrees. Now raise the tilt to 90.0°. The eccentricity reads 1.414, against 1.155 for the same tilt on a cone of half-angle 30°: the wider cone gives the more open hyperbola.

The figure draws a right cone. Apollonius' cone may be oblique, and his rules for the latus rectum hold for any circular cone. The cone is drawn only to 4 units above and below the apex, so long sections run off its edge, and the plane always turns about the same tangent line, which is one choice among many.

Sources: T. L. Heath, Apollonius of Perga: Treatise on Conic Sections (Cambridge, 1896), archive.org/details/treatiseonconics00apolrich, p. lxviii: 'We possess only the most meagre information about Apollonius', with the reigns of Ptolemy Euergetes (247 to 222 BC) and Ptolemy Philopator; Geminus, quoted by Eutocius, on the 'great geometer' (Reported); 'Seven Books only out of the eight have survived, four in the original Greek, and three in an Arabic translation.' Read 2026-10-08.; On the birth year: MacTutor, Apollonius of Perga (mathshistory.st-andrews.ac.uk/Biographies/Apollonius/) gives 'about 262 BC', and Heath's History of Greek Mathematics (1921, vol. II p. 126) says 'probably born about 262 B.C.', and moves his prime to Euergetes: 'he flourished ... in the reign of Ptolemy Euergetes (247-222 B.C.)'. That revises Heath's 1896 account, in which he was born under Euergetes and flourished under Philopator. Heath's two accounts disagree, so no year is given here as secure. Grade: Secondary. Read 2026-10-08.; Heath 1896, Preface, pp. ix to x: Heath first wrote a literal translation, then rewrote the work with 'a new and uniform notation', diminishing its bulk 'by considerably more than one-half'. Quotations from his main text are therefore his paraphrase, not Apollonius' words. Read 2026-10-08.; Heath 1896, p. 1 ('The Cone'): 'a straight line indefinite in length, and passing always through a fixed point' moved round a circle not in its plane traces 'the surface of a double cone'; the cone is scalene 'except in the particular case where the axis is perpendicular to the base'. Heath's paraphrase of Conics I, definitions 1 to 3 (J. L. Heiberg's edition, vol. 1, Leipzig, 1891, Greek p. 6, Latin p. 7, archive.org/details/apolloniipergaei01apoluoft). Grade: Paraphrase. Read 2026-10-08.; Heath 1896, p. 13, his Prop. 4 [I.14], on opposite branches (Paraphrase); Heiberg vol. 1 p. 53: 'uocentur autem tales sectiones oppositae' (Attested). Heath 1896 p. clxi: 'hyperbola (which last means only one branch of a hyperbola)'. Read 2026-10-08.; Heath 1896, p. lxx: Apollonius' preface to Book I, from what Heath calls 'a literal translation of the dedicatory letters' (p. lxix). Grade: Attested, in translation. Read 2026-10-08.; Heath 1896, p. xxiv: Eutocius, quoting Geminus with approval, on right cones cut 'by a plane perpendicular to one of the generating lines' and on the old names. Grade: Reported. Read 2026-10-08.; T. L. Heath, A History of Greek Mathematics (Oxford, 1921), vol. I, archive.org/details/historyofgreekm01heat, p. 251: Menaechmus 'was a pupil of Eudoxus and flourished about the middle of the fourth century B.C.', and his discovery of the conics is inferred from his solutions and the epigram; p. 260: Eratosthenes' epigram, 'or to cut the cone in the triads of Menaechmus' (Attested, in Heath's translation). Vol. II, archive.org/details/historyofgreekm02heat, p. 110: 'On this we have no information whatever.' Read 2026-10-08.; Heath 1896, p. lxxvii: Eutocius, 'in every cone, whether right or scalene, all the sections are found' (Reported); pp. lxxvii to lxxviii: Euclid's cone '(presumably right)' or cylinder; 'Archimedes states expressly that all sections of a cone which meet all the generators (and here the cone may be oblique) are either circles or "sections of an acute-angled cone"', which could only have been found, Heath argues, by 'a method which would equally show that hyperbolic and parabolic sections could be produced in the same general manner'; the claim 'rests on a misapprehension', and 'Apollonius was making an entirely new departure' (Secondary); p. xxxvi: Euclid's ellipse from a cone or a cylinder, 'assuming the section to lie wholly between the apex of the cone and its base'. Heath 1921, vol. II p. 126: Apollonius was the first 'to base the theory of conics on the production of all three in the most general way from any kind of circular cone, right or oblique'. Read 2026-10-08.; Wikipedia, Conic section (rev. 1377246756), citing Thomas and Finney 1979, p. 434: with β the angle between the cone's surface and its axis and α the angle between the cutting plane and the axis, 'the eccentricity is' cos α ÷ cos β. A plane tilted t from the base makes 90° − t with the axis, so cos α = sin(tilt). Grade: Secondary. Read 2026-10-08.; Computed here with Node on 2026-10-08 from e = sin(tilt) ÷ cos(half-angle): 0.577 (tilt 30°), 0.742 (40°), 1.005 (60.5°), 1.150 (85°) and 1.155 (90°) on a half-angle of 30°, and 1.414 for a tilt of 90° on a half-angle of 45°. The parabola falls at tilt = 90° − half-angle.

Check yourself · 4 cards · spaced review in the study dashboard

  1. recall

    Before Apollonius, how did geometers obtain the three conic sections from cones, and what did they call the curves?

    Show answer

    According to Eutocius, quoting Geminus, they used only right cones, sorted by the angle at the apex: less than, equal to or greater than a right angle. Each kind was cut by a plane perpendicular to one of its generating lines and gave one curve: the section of an acute-angled cone (the ellipse), of a right-angled cone (the parabola) and of an obtuse-angled cone (the hyperbola).

  2. explain

    In the cone figure, how does the tilt of the cutting plane, compared with the slope of the cone's side, decide which section is cut?

    Show answer

    A level plane cuts a circle. A plane shallower than the cone's side closes the curve into an ellipse, and a plane parallel to the side gives a parabola. A steeper plane also cuts the other half of the double cone and gives a hyperbola with two branches, which Apollonius calls opposite sections.

  3. apply

    A double cone has a half-angle of 30°, and the eccentricity of a section is sin(tilt) ÷ cos(half-angle). What section does a plane tilted at 40° cut, what is its eccentricity, and at what tilt does the section become a parabola?

    Show answer

    An ellipse: sin 40° ÷ cos 30° = 0.643 ÷ 0.866 = 0.742, and the plane is shallower than the cone's side by 20°, since 40 − (90 − 30) = −20. The parabola comes at a tilt of 60°, where the tilt and the half-angle add up to 90°.

  4. explain

    Eutocius credits Apollonius with showing that all the sections are found in every cone, right or scalene. What is Heath's verdict on that claim?

    Show answer

    Heath says the claim to priority 'rests on a misapprehension'. Euclid knew that a cone (presumably right, Heath says) or a cylinder, cut by a plane not parallel to the base, gives an ellipse when the section lies wholly between the apex and the base. Archimedes states that every section of a cone, even an oblique one, that meets all its generating lines is a circle or an ellipse, and Heath infers that the same method would give the parabola and the hyperbola. Yet he grants that 'in the generality of his treatment of the subject from the very beginning, Apollonius was making an entirely new departure'.

II · The parabola

The parabola: an area laid along exactly.

Conics I.11 takes a section whose diameter is parallel to one side of the axial triangle, the triangle a plane through the cone's axis cuts out. Apollonius proves that the square on any ordinate equals a rectangle laid along a fixed line, with the abscissa as its width. In Heath's letters P is the vertex, Q a point on the curve, QV the ordinate (half of a chord that the diameter cuts in two), PV the abscissa, and PL the fixed line, drawn at right angles to the diameter at P: QV² = PL·PV. Heath's paraphrase ends 'Hence the section is called a Parabola'. Apollonius' own words, in Heiberg's Latin, are 'uocetur autem talis sectio parabola', let such a section be called a parabola. He calls PL the latus rectum, which Heath renders 'the erect side'. Like Elements I.47, where Euclid sets the square on the side opposite the right angle equal to the squares on the other two sides (see the Academy on I.47), the property is an equality of areas. In modern notation, with PL = p, PV = x and QV = y, the property is y² = px.

The figure lays the property out with Heath's letters. The axis runs across with P at the left; Q moves on the upper half of the curve, V is the foot of its ordinate, and PL hangs down from P. The square on QV stands on QV, above the axis. Below the axis, the rectangle on PV with height PL has the same fill, because the two are equal. A faint unit grid lets you count both in whole-number cases. Two sliders set the Latus rectum PL (1 to 8) and the Abscissa PV (0 to 10). You can also drag Q along the curve, and Move Q sweeps it back and forth. The readouts give the Abscissa PV, the Ordinate QV, the Square on QV and the Rectangle PV × PL; the last two are always equal. The square grows in step with PV, so its side, the ordinate, grows only as the square root: doubling the ordinate takes four times the abscissa.

The name is borrowed from an older method. To apply an area to a line is to build a parallelogram equal to that area with the line as one side (D. E. Joyce on Euclid, Elements I.44). Proclus, commenting on I.44, reports Eudemus as saying that the application of areas, with its exceeding and falling short, is ancient, one of the 'discoveries of the Muse of the Pythagoreans'. In the same passage Proclus says that later geometers (Heath adds: Apollonius) gave these names to the conic lines: parabola (application), hyperbola (exceeding) and ellipse (falling short). For the parabola the rectangle fits the latus rectum exactly. Heath, writing in 1896, says that 'All authorities agree' in giving Apollonius the three names, and Archimedes still wrote 'section of a right-angled cone'. Heath could not weigh Diocles' On Burning Mirrors, found in Arabic translation and published only in 1976. From it G. J. Toomer argued instead that 'parabola' and 'hyperbola' are older than Apollonius; MacTutor's page on Diocles adds 'ellipse'. The sources used here do not settle it. Archimedes himself proved that a segment of a parabola is four-thirds of the triangle with the same base and height, first by his mechanical method and then by pure geometry, summing 1 + 1/4 + 1/16 + ... of inscribed triangles (Quadrature of the Parabola). The Academy has his sphere and cylinder.

Try this

  • Leave the Latus rectum PL at 4 and the Abscissa PV at 1. The Ordinate QV reads 2.00, and the Square on QV and the Rectangle PV × PL both read 4.00 sq units: on the grid, a 2 by 2 square and a 1 by 4 rectangle.
  • Set the Abscissa PV to 4, then 9. The ordinate reads 4.00, then 6.00, and both areas read 16.00, then 36.00 sq units. Four times the abscissa gives only twice the ordinate.
  • Set PV to 0.25, 2.25 and 6.25. The ordinate reads 1.00, 3.00 and 5.00, and the areas 1.00, 9.00 and 25.00 sq units. With PL at 4, the ordinate is a whole number whenever 4 × PV is a square number.

Apollonius proves the property for a diameter that comes out of his cone construction, which in general is not the axis, so his ordinates usually meet the diameter at a slant. The figure uses the axis, where they meet at right angles. The equation y² = px is Heath's modern shorthand; Apollonius states the property as an equality of areas.

Sources: Heath 1896, pp. 8 to 9, his Prop. 1 [I.11]: the square on any ordinate 'is equal to a rectangle applied ... to the fixed straight line PL drawn at right angles to PM with altitude equal to the corresponding abscissa PV. Hence the section is called a Parabola.' An ordinate is half a chord bisected by the diameter (p. 8). Grade: Paraphrase. J. L. Heiberg, Apollonii Pergaei quae Graece exstant, vol. 1 (Leipzig, 1891), archive.org/details/apolloniipergaei01apoluoft, Latin p. 39: 'uocetur autem talis sectio parabola'; p. 43, of the line PL: 'uocetur autem etiam latus rectum'. Grade: Attested, in Heiberg's Latin; the English renderings are made here. Read 2026-10-08.; Heath 1896, p. 11 and appendix p. clxii: the parameter is 'the latus rectum (i.e. the erect side ...)'. p. lxxix: 'All authorities agree in attributing to Apollonius the designation of the three conics by the names parabola, ellipse and hyperbola'. p. xxiv: 'The sections are so described by Archimedes.' pp. lxxviii to lxxix: Apollonius' original diameter 'is not in general one of the principal diameters'. Read 2026-10-08.; D. E. Joyce, Euclid's Elements, Book I Proposition 44 and its Guide, mathcs.clarku.edu/~djoyce/elements/bookI/propI44.html: 'To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle' (Attested, in translation); to apply an area to a line means 'to construct a parallelogram equal to that area with one side as the given line and one angle equal to the given angle'. Read 2026-10-08.; Proclus, Commentary on Euclid I, on I.44, translated in T. L. Heath, The Thirteen Books of Euclid's Elements, vol. I (Cambridge, 1908), archive.org/details/thirteenbookseu02heibgoog, p. 343: 'These things, says Eudemus ..., are ancient and are discoveries of the Muse of the Pythagoreans'; 'It was from the Pythagoreans that later geometers [i.e. Apollonius] took the names', the bracket being Heath's. Grade: Reported (Proclus, 5th century AD, quoting Eudemus, 4th century BC). Proclus's Greek and Friedlein's edition were not checked. Read 2026-10-08.; MacTutor, Menaechmus, mathshistory.st-andrews.ac.uk/Biographies/Menaechmus/: 'recent evidence in Diocles' On burning mirrors discovered in Arabic translation in the 1970s, led G J Toomer to claim that both the names "parabola" and "hyperbola" are older than Apollonius'. MacTutor, Diocles, mathshistory.st-andrews.ac.uk/Biographies/Diocles/: Toomer published the translation in 1976, and this page includes 'ellipse'; also 'No writing of Diocles was known to Heath in 1921'. Toomer's edition was not read. Read 2026-10-08.; Heath 1921, vol. II pp. 21 to 22: the quadrature of the parabola was 'the very first theorem which he found out by means of mechanics', the segment being 'four-thirds of that of the triangle which has the same base and height' (Paraphrase of Archimedes, The Method); p. 86: Props. 1 to 17 by mechanics confirmed by exhaustion, Props. 18 to 24 by pure geometry; p. 90: the series 1 + 1/4 + (1/4)² + ... (Prop. 22). Read 2026-10-08.; Computed here with Node on 2026-10-08 from QV² = PL × PV with PL = 4: ordinates 1, 2, 3, 4, 5 and 6 at abscissas 0.25, 1, 2.25, 4, 6.25 and 9.

Check yourself · 4 cards · spaced review in the study dashboard

  1. recall

    In Heath's letters, what equality does Conics I.11 prove for the parabola, and what are P, Q, V and PL?

    Show answer

    QV² = PL·PV: the square on the ordinate QV equals the rectangle on the fixed line PL whose width is the abscissa PV. P is the vertex, Q a point on the curve and V the foot of its ordinate on the diameter. PL, the latus rectum, is drawn at right angles to the diameter at P.

  2. apply

    A parabola has latus rectum PL = 4. How long is the ordinate QV at abscissas PV = 1, 4 and 9, and what does the pattern show?

    Show answer

    QV² = 4 × PV gives squares of 4, 16 and 36 square units, so QV is 2, 4 and 6. Four times the abscissa, from 1 to 4, only doubles the ordinate, from 2 to 4: the ordinate grows as the square root of the abscissa.

  3. explain

    What does it mean to apply an area to a line, and why does that suit the name 'parabola'?

    Show answer

    To apply an area to a line is to build a parallelogram equal to that area with the line as one side (D. E. Joyce on Euclid, Elements I.44). In the parabola the rectangle equal to the square on the ordinate, laid along the latus rectum, fits it exactly, neither falling short nor exceeding. Proclus, citing Eudemus, says the application of areas, with its exceeding and falling short, was an ancient Pythagorean discovery.

  4. recall

    Who gave the parabola its name? State the two positions on this page, and whether the question is settled.

    Show answer

    Heath, writing in 1896, says that 'All authorities agree' in giving Apollonius the names parabola, ellipse and hyperbola, and Archimedes still wrote 'section of a right-angled cone'. G. J. Toomer, working from Diocles' On Burning Mirrors, published only in 1976, argued that 'parabola' and 'hyperbola' are older than Apollonius. The sources used on the page do not settle it.

III · The ellipse

The ellipse: an area that falls short.

Conics I.13 gives the ellipse. The square on the ordinate again equals a rectangle laid along the latus rectum with the abscissa as its width, but now that rectangle falls short of the full rectangle on PV and PL. Apollonius names the shortfall exactly. It is a figure similar and similarly placed to the rectangle on the diameter PP′ with height PL, which he calls the figure (eidos) of the section; PP′ is its transverse side and PL its erect side. Heath's paraphrase ends 'The section is therefore called an Ellipse'. Heiberg's Latin of Apollonius has 'et figura deficiens simili similiterque posita rectangulo a diametro parametroque comprehenso; uocetur autem talis sectio ellipsis': falling short by a figure similar and similarly placed to the rectangle contained by the diameter and the parameter, and let such a section be called an ellipse. With PP′ = d, the property is y² = px − (p/d)x². For the ellipse and the hyperbola Heath contrasts Apollonius' form, 'the equality of two areas', with Archimedes', 'the equality of two proportions'; for the parabola, he notes, Archimedes too uses the latus rectum. Pappus later described the shortfall as a square. Heath calls this 'some confusion', since in Apollonius it is a rectangle similar to the figure.

The figure keeps the parabola's layout and adds P′, the other end of the diameter, d units to the right of P. A guide line runs from P′ to L, and its depth below V gives the height of the applied rectangle, PL × (1 − PV ÷ PP′). That rectangle has the same fill as the square on QV, because the two are equal. The rest of the full rectangle PV × PL, hatched, is the shortfall: PV wide and PL × PV ÷ PP′ high, so its sides are in the ratio of PL to PP′ and it is similar to the figure. Three sliders set the Diameter PP′ (2 to 12), the Latus rectum PL (0.5 to 12) and the Abscissa PV (0 up to PP′). Q drags along the curve, and Move Q sweeps it. The readouts give the Abscissa PV, the Square on QV, the Applied rectangle and Falls short by. While PL is shorter than PP′, PP′ is the long axis. When PL is longer, PP′ is the short axis, and when the two are equal the curve is a circle.

Euclid had already set the matching problem. Elements VI.28 applies to a given line a parallelogram equal to a given figure 'but falling short by a parallelogram similar to a given one', and adds a limit: the given figure 'must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram'. In modern algebra, with a square as the given parallelogram, it solves ax − x² = C (Joyce), and Heath calls the general problem 'equivalent to that of solving geometrically a mixed quadratic equation'. Finding the abscissa for a given square on QV is a problem of this kind, with PL as the line and the figure as the given parallelogram, and the limit shows in the figure. The square on QV is largest when PV is half of PP′. There the applied rectangle is PP′ ÷ 2 wide and PL ÷ 2 high: it stands on half of PL, it is similar to the figure, and its area, PL × PP′ ÷ 4, is exactly Euclid's limit. Every smaller value occurs at two abscissas. Euclid still called the curve the section of an acute-angled cone, and knew it could be cut from a cylinder as well as from a cone. For the ellipse as an orbit, see equal areas in the Astronomy lab. For a reading plan through Euclid and Apollonius, see the Library's Geometry, Euclid and Trigonometry.

Try this

  • At the defaults (Diameter PP′ 9, Latus rectum PL 4.5, Abscissa PV 6), the Square on QV and the Applied rectangle both read 9.00 sq units, a rectangle 6 by 1.5, and Falls short by reads 18.00, a rectangle 6 by 3. Together they fill the full rectangle PV × PL, 6 × 4.5 = 27.
  • Set PV to 3: the square and the applied rectangle read 9.00 again, now 3 by 3, and the shortfall 4.50. Set PV to 1, then 8: both read 4.00, with shortfalls of 0.50 and 32.00. Each value of the square occurs at two abscissas, placed symmetrically about the middle of PP′.
  • Set PP′ and PL both to 10, and the curve becomes a circle. At PV 1, 2 and 5 the Square on QV reads 9.00, 16.00 and 25.00 sq units, and the applied rectangle is 1 by 9, 2 by 8 and 5 by 5: the square on the ordinate equals the rectangle on the two parts of the diameter.

As in figure II the diameter is an axis, so the ordinates meet it at right angles; Apollonius' diameters are in general oblique. The rectangles sit below the axis, where the lower half of the curve also runs, so that half is drawn faint.

Sources: Heath 1896, pp. 11 to 12, his Prop. 3 [I.13]: 'Thus the square on the ordinate is equal to a rectangle whose height is equal to the abscissa and whose base lies along the fixed straight line PL but falls short of it ... by a length equal to the difference between VR and PL. The section is therefore called an Ellipse.' His footnote gives y² = px − (p/d)x². Grade: Paraphrase. Heiberg vol. 1 (1891), Latin p. 49, as quoted. Grade: Attested, in Heiberg's Latin; the English rendering is made here. Read 2026-10-08.; Heath 1896, p. 11 and appendix p. clxii: the latus rectum and the diameter are the erect side and the transverse side 'of the figure (εἶδος) on, or applied to, the diameter ..., i.e. of the rectangle contained by PL, PP′ as drawn'. Read 2026-10-08.; Heath 1908, vol. I p. 345: the rectangle exceeds or falls short 'by a figure similar and similarly situated to the rectangle contained by the given diameter and p', so 'Apollonius' nomenclature followed exactly the traditional theory of application, exceeding, and falling-short'. Read 2026-10-08.; Heath 1896, pp. lxxx to lxxxi: 'In Archimedes, on the other hand, while the parameter duly appears with reference to the parabola, no such line is anywhere mentioned in connexion with the ellipse or hyperbola'; 'Thus Apollonius' equation expressed the equality of two areas, while Archimedes' equation expressed the equality of two proportions'. Grade: Secondary. Read 2026-10-08.; Heath 1896, pp. lxxxiii to lxxxiv, on Pappus (Hultsch p. 674): 'There is evidently some confusion here, because in the definitions of Apollonius there is no question of exceeding or falling-short by a square'. Grade: Doubtful, for Pappus's description. Read 2026-10-08.; D. E. Joyce, Euclid's Elements, Book VI Proposition 28 and its Guide, mathcs.clarku.edu/~djoyce/elements/bookVI/propVI28.html: the enunciation as quoted (Attested, in translation); the Guide reads the construction, for rectangles, as solving the quadratic ax − x² = C (a modern reading, not Euclid's language). Read 2026-10-08.; Heath 1908, vol. I p. 344: 'the general problem here stated is equivalent to that of solving geometrically a mixed quadratic equation.' Read 2026-10-08.; Heath 1896, p. xxxvi: 'Euclid still used the old names for the three conic sections, but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane not parallel to the base (assuming the section to lie wholly between the apex of the cone and its base), and also by cutting a cylinder.' Read 2026-10-08.; Computed here with Node on 2026-10-08 from y² = px − (p/d)x²: the squares, applied rectangles and shortfalls in the try prompts (d 9 and p 4.5; d = p = 10). The match with VI.28, with the largest square, pd/4, at PV = d/2, is worked out here.

Check yourself · 4 cards · spaced review in the study dashboard

  1. recall

    In Conics I.13, by what does the rectangle equal to the square on the ordinate fall short, and what does Apollonius call the rectangle it is compared with?

    Show answer

    It falls short of the full rectangle on PV and PL by a figure similar and similarly placed to the rectangle on the diameter PP′ with height PL. Apollonius calls that rectangle the figure (eidos) of the section; PP′ is its transverse side and PL its erect side.

  2. apply

    An ellipse has diameter PP′ = 9 and latus rectum PL = 4.5. At abscissa PV = 6, what are the height of the applied rectangle, the square on QV and the shortfall?

    Show answer

    The height is PL × (1 − PV ÷ PP′) = 4.5 × (1 − 6 ÷ 9) = 1.5, so the square on QV is 6 × 1.5 = 9 square units. The shortfall is 6 wide and 4.5 × 6 ÷ 9 = 3 high, 18 square units. Together they fill the full rectangle PV × PL, 6 × 4.5 = 27.

  3. connect

    Where does the square on QV in an ellipse reach its largest value, and which limit in Euclid's Elements VI.28 matches it?

    Show answer

    The square on QV is largest when PV is half of PP′, and every smaller value occurs at two abscissas, placed symmetrically about the middle of PP′. VI.28 applies to a line a parallelogram that falls short by one similar to a given one, and adds that the given figure 'must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram'. At PV = PP′ ÷ 2 the applied rectangle is PP′ ÷ 2 by PL ÷ 2, the parallelogram on half of PL similar to the figure, so the largest square, PL × PP′ ÷ 4, is exactly that limit.

  4. apply

    In the ellipse figure, set the diameter PP′ and the latus rectum PL both to 10. What curve results, and what does the square on QV equal at PV = 2?

    Show answer

    A circle, because PL equals PP′. At PV = 2 the square on QV is 16 square units, equal to the applied rectangle 2 by 8: the square on the ordinate equals the rectangle on the two parts of the diameter, 2 and 8.

IV · The hyperbola

The hyperbola: an area that overshoots.

Conics I.12 gives the hyperbola. Here the rectangle laid along the latus rectum, with the abscissa as its width, overlaps the full rectangle on PV and PL, and exceeds it by a figure similar and similarly placed to the one on the transverse diameter PP′ and the latus rectum. Heath's paraphrase ends 'Hence the section is called a Hyperbola'. Heiberg's Latin of Apollonius has 'excedens figura simili similiterque posita ... uocetur autem talis sectio hyperbola', exceeding by a similar and similarly placed figure, and let such a section be called a hyperbola. With PP′ = d, the property is y² = px + (p/d)x². The other end of the diameter, P′, belongs to the opposite section. In I.14 the two opposite sections have the same diameter and equal latera recta, and their common transverse side is the line between their vertices. Heath notes that Apollonius treats the pair as one curve from I.16 on, yet 'continues throughout to speak of them as two independent curves' and proves each proposition for them separately.

The figure is the ellipse's figure with P′ moved to the other side: d units to the left of P, at the vertex of the opposite section. When P′ fits in the frame it is drawn, with the outlined figure PP′ × PL and a faint arc of the opposite section through it; when it does not, a small note points the way. The guide line from P′ runs on through L, so below V it lies deeper than PL: the applied rectangle has height PL × (1 + PV ÷ PP′) and reaches below the full rectangle PV × PL. The hatched excess, PV wide and PL × PV ÷ PP′ high, is similar to the figure, as Apollonius says. Sliders set the Diameter PP′ (1 to 12), the Latus rectum PL (0.5 to 12) and the Abscissa PV (0 to 10). Q drags along the curve, and Move Q sweeps it. The readouts are Abscissa PV, Square on QV, Applied rectangle and Exceeds by. However far Q goes, the curve never closes.

Euclid's Elements VI.29 is the matching construction, a parallelogram applied to a line 'but exceeding it by a parallelogram similar to a given one'. Unlike VI.28 it needs no limit, and when the given parallelogram is a square it solves, in modern algebra, ax + x² = C (Joyce). The curve was in use long before Apollonius wrote. Heath reports that Eutocius describes two solutions by Menaechmus to the problem of two mean proportionals, on which the doubling of the cube depends: one finds the point where two parabolas meet, the other the point where a parabola meets a rectangular hyperbola. Apollonius adds the lines the hyperbola approaches. In Conics II.1 he marks off, on the tangent at P, equal lengths on each side whose square is one-fourth of the figure, and proves that the lines from the centre through their ends do not meet the curve: 'ergo ΓΔ, ΓΕ asymptotae sectionis sunt', therefore they are the asymptotes of the section. Opposite sections share them (II.15). Figure VI draws them for eccentricities above 1, and section V shows the hyperbola's two foci at work in telescope mirrors.

Try this

  • At the defaults (Diameter PP′ 7, Latus rectum PL 3.5, Abscissa PV 2), the Square on QV and the Applied rectangle read 9.00 sq units, a rectangle 2 by 4.5, and Exceeds by reads 2.00, a rectangle 2 by 1. The full rectangle PV × PL is only 7.
  • Set PV to 1, then 7. The square reads 4.00, then 49.00, and the excess 0.50, then 24.50. At PV 7, equal to PP′, the applied height has doubled to 7, so the excess equals the whole rectangle PV × PL, 7 × 3.5 = 24.5.
  • Set PP′ to 4, PL to 2 and PV to 4: the Square on QV reads 16.00 sq units, a 4 by 4 square equal to an applied rectangle 4 by 4. Halve all three (PP′ 2, PL 1, PV 2) and it reads 4.00, a quarter as much, because every length in the figure has halved.

The figure shows the branch through P, with its lower half faint, and of the opposite section only a faint arc through P′, drawn when P′ fits in the frame. As in figures II and III, the diameter is an axis with ordinates at right angles, and the asymptotes are not drawn here.

Sources: Heath 1896, pp. 9 to 10, his Prop. 2 [I.12]: 'It follows that the square on the ordinate is equal to a rectangle whose height is equal to the abscissa and whose base lies along the fixed straight line PL but overlaps ... it by a length equal to the difference between VR and PL. Hence the section is called a Hyperbola.' His footnote gives y² = px + (p/d)x², referred to a diameter and the tangent at its end. Grade: Paraphrase. Heiberg vol. 1 (1891), Latin p. 43: 'excedens figura simili similiterque posita rectangulo comprehenso recta sub angulo trianguli extrinsecus posito subtendenti parametroque; uocetur autem talis sectio hyperbola.' Grade: Attested, in Heiberg's Latin; the English rendering is made here. Read 2026-10-08.; Heath 1896, p. 13, his Prop. 4 [I.14] (Paraphrase); Heiberg vol. 1 p. 53: 'et transuersum figurae latus commune recta inter uertices sectionum posita' (Attested). Heath 1896 p. lxxxiv: the opposite branches are first regarded as one curve in I.16 [Prop. 6], yet he 'continues throughout to speak of them as two independent curves'. Read 2026-10-08.; D. E. Joyce, Euclid's Elements, Book VI Proposition 29 and its Guide, mathcs.clarku.edu/~djoyce/elements/bookVI/propVI29.html: 'To apply a parallelogram equal to a given rectilinear figure to a given straight line but exceeding it by a parallelogram similar to a given one' (Attested, in translation); 'it solves the quadratic equation ax + x² = C' (Joyce's modern reading, for the case 'when the given parallelogram D is a square'). Read 2026-10-08.; Heath 1921, vol. I p. 251: 'Two solutions by Menaechmus of the problem of finding two mean proportionals are described by Eutocius; both find a certain point as the intersection between two conics, in the one case two parabolas, in the other a parabola and a rectangular hyperbola.' Grade: Secondary (Heath's summary; Eutocius was not read). Read 2026-10-08.; Conics II.1 and II.15 in Heiberg vol. 1 (1891), Latin pp. 193 and 195 ('rectae a centro sectionis ad terminos sumptos contingentis ductae cum sectione non concurrent'; 'ergo ΓΔ, ΓΕ asymptotae sectionis sunt') and p. 219 ('Sectionum oppositarum communes sunt asymptotae', read from the text layer; the page image was not viewed). Grade: Attested, in Heiberg's Latin; Heath 1896 p. 53, his Prop. 28, paraphrases them together. Read 2026-10-08.; Computed here with Node on 2026-10-08 from y² = px + (p/d)x²: the squares, applied rectangles and excesses in the try prompts (d 7 and p 3.5; d 4 and p 2; d 2 and p 1).

Check yourself · 4 cards · spaced review in the study dashboard

  1. recall

    In Conics I.12, how does the rectangle equal to the square on the ordinate compare with the full rectangle on PV and PL?

    Show answer

    It overlaps the full rectangle and exceeds it by a figure similar and similarly placed to the rectangle on the transverse diameter PP′ and the latus rectum PL. Apollonius names the section from this excess, and with PP′ = d and PL = p the property is y² = px + (p/d)x².

  2. apply

    A hyperbola has diameter PP′ = 7 and latus rectum PL = 3.5. At abscissa PV = 2, what are the height of the applied rectangle, the square on QV and the excess?

    Show answer

    The height is PL × (1 + PV ÷ PP′) = 3.5 × (1 + 2 ÷ 7) = 4.5, so the square on QV is 2 × 4.5 = 9 square units. The excess is 2 wide and 3.5 × 2 ÷ 7 = 1 high, 2 square units, beyond the full rectangle PV × PL of 7.

  3. connect

    Which construction in Euclid's Elements matches the hyperbola's area property, and how does it differ from VI.28?

    Show answer

    Elements VI.29, which applies a parallelogram to a line 'but exceeding it by a parallelogram similar to a given one'. Unlike VI.28 it needs no limit. When the given parallelogram is a square, it solves ax + x² = C in modern algebra.

  4. recall

    How does Apollonius obtain the asymptotes of a hyperbola in Conics II.1, and which curves share them?

    Show answer

    On the tangent at P he marks off equal lengths on each side whose square is one-fourth of the figure, and proves that the lines from the centre through their ends do not meet the curve. The opposite sections share the same asymptotes (II.15).

V · The foci

Two points that steer the curve.

Apollonius has no word for a focus. In Book III (from III.45) he fixes two points on the axis of an ellipse or hyperbola by another application of areas: for each point, the rectangle formed by its distances to the two vertices equals one-fourth of the figure. For the ellipse the points lie between the vertices; for the hyperbola they lie within each branch, on the axis produced. He calls them 'the points arising out of the application'. He proves that the lines from them to any point of the curve make equal angles with the tangent there (III.48). He also proves that the difference of those two lines, for the hyperbola (III.51), and their sum, for the ellipse (III.52), equals the axis. Read as a mirror, equal angles mean that in the ellipse a ray from one point reflects towards the other, and in the hyperbola away from it, as if the other had sent it. The mirror reading is Heath's, who thought it 'certain that Apollonius was aware' of it for the ellipse; Apollonius states only the equal angles. Heath is plain about the parabola: 'The focus of a parabola is not used or mentioned by Apollonius.' Its focus has other witnesses. MacTutor, following G. J. Toomer, credits Diocles with the first proof of the focal property of a parabolic mirror, in On Burning Mirrors; his date is disputed, about 240 to 180 BC in MacTutor and a century or more after Apollonius in Heath. A Greek fragment that Heath dates probably no later than Apollonius puts the burning point a quarter of the latus rectum from the vertex.

The figure has three modes, chosen with the buttons Ellipse, Hyperbola and Parabola. The slider Point P moves P along the curve, and you can drag P. The focal lines PF₁ and PF₂ are drawn with the tangent at P, and two arcs mark the equal angles. The ellipse is 10 units long and 6 high, with foci 4 from the centre. Its readouts are PF₁, PF₂, PF₁ + PF₂ and Angles with the tangent, and Show the string draws the two focal lines as one taut string pinned at F₁ and F₂. The hyperbola has its vertices 6 apart and its foci 5 from the centre. Its readouts are PF₁, PF₂, PF₂ − PF₁ and Angles with the tangent, and P stays on the branch round F₁. The parabola has latus rectum 4, so F lies 1 from the vertex A. Its readouts are PF, AN + ¼ latus rectum and Angles with the tangent, where N is the foot of the ordinate from P. Send light from F₁ (Send light from F for the parabola) releases about 24 rays at once and leaves their paths as faint trails. From the ellipse they all reach F₂ at the same moment, since every path is 10 units long. From the hyperbola they leave along lines that run back, dashed, to F₂, as if F₂ had sent them. From the parabola they leave parallel to the axis, side by side like a flat wavefront.

The name came in 1604. Kepler, in Ad Vitellionem paralipomena, says the points have a definite definition but 'nomen nullum', no name, and supplies one: 'Nos lucis causa et oculis in mechanicam intentis ea puncta focos appellabimus', for the sake of light and with our eyes on mechanics we shall call these points foci. Focus is Latin for a hearth, but that is not the reason Kepler gives. Fire is close to the naming all the same. A page earlier Kepler recalls what Witelo wrote 'de speculo parabolico adurente', on the burning parabolic mirror, and the same section closes by calling the study of the conic sections necessary 'ad ignes incendendos', for kindling fires. Replying in January 1605 to his correspondent J. G. Brengger, Kepler wrote that the true focus of a parabolic mirror is where all rays parallel to the axis gather, 'atque hic demum est violentissima incensio', and just here is the fiercest burning. He also gave the parabola a second, 'blind' focus, imagined on the axis at an infinite distance. The same properties do practical work now. In an electrohydraulic lithotripter a shock wave is made at one focus of a half-ellipsoid reflector and focused on a kidney stone at the other, though not every lithotripter uses an ellipse. Parabolic mirrors gather light in reflecting telescopes and send out the beams of searchlights and car headlights. A hyperbolic mirror sends rays aimed at one of its foci to the other, and both of the Hubble Space Telescope's mirrors are hyperbolic. In 1822 G. P. Dandelin showed that the foci of a section are the points where the cutting plane touches the spheres inscribed in the cone, which ties this figure back to the cone.

Try this

  • In Ellipse mode P starts at the top of the curve: PF₁ and PF₂ read 5.00 each, PF₁ + PF₂ reads 10.00, and Angles with the tangent reads 36.9° and 36.9°. Move P to the right-hand end of the long axis: PF₁ reads 9.00 and PF₂ 1.00, the sum is still 10.00, and the angles are 90.0° and 90.0°. There F₂ is 1 from this vertex and, by symmetry, 9 from the other, and 1 × 9 = 9, one-fourth of the figure 10 × 3.6: Apollonius' rule for fixing the points.
  • Press Hyperbola and move P along its branch. PF₂ − PF₁ stays at 6.00, the distance between the vertices. At the vertex PF₁ reads 2.00 and PF₂ 8.00, so F₁ is 2 from this vertex and, by symmetry, 8 from the other, and 2 × 8 = 16, one-fourth of this figure (axis 6, latus rectum 32 ÷ 3, about 10.67). Move P out along the branch until PF₁ reads 10.00: PF₂ reads 16.00, and the equal angles have closed from 90.0° to 18.4° and 18.4°.
  • Press Parabola, then Send light from F. The rays leave the focus together and come off the curve parallel to the axis. Move P: PF and AN + ¼ latus rectum read the same everywhere, 1.00 at the vertex, and with P straight above F both read 2.00 and the angles are 45.0° and 45.0°.

The rays are ideal straight lines, reflected without loss by a perfect curve, and everything happens in one plane. The parabola mode is not in Apollonius, who never mentions the parabola's focus.

Sources: Heath 1896, pp. 113 to 114: 'The foci are not spoken of by Apollonius under any equivalent of that name, but they are determined as the two points on the axis of a central conic (lying in the case of the ellipse between the vertices, and in the case of the hyperbola within each branch, or on the axis produced) such that the rectangles AS.SA′, AS′.S′A′ are each equal to "one-fourth part of the figure of the conic," i.e. ¼pₐ.AA′ or CB²' (Secondary); Apollonius' phrase τὰ ἐκ τῆς παραβολῆς γινόμενα σημεῖα, 'the points arising out of the application' (Attested, in Heath's translation); p. 114: 'The focus of a parabola is not used or mentioned by Apollonius.' Read 2026-10-08.; Heath 1896, p. 116, his Prop. 71 [III.48]: 'The focal distances of P make equal angles with the tangent at that point' (Paraphrase); Heiberg vol. 1 (1891), Latin p. 431: 'rectas a puncto contactus ad puncta adplicatione orta ductas ad contingentem angulos aequales efficere' (Attested). Read 2026-10-08.; Heath 1896, p. 118, his Prop. 73 [III.51, 52], the sum and the difference together (Paraphrase); Heiberg vol. 1, Latin p. 435, III.51 for the hyperbola, 'maior minorem excedit axe', and p. 437, III.52 for the ellipse, 'eae axi aequales erunt' (Attested). Read 2026-10-08.; Heath 1921, vol. II p. 200: 'it is certain that Apollonius was aware that an ellipse has the property of reflecting all rays through one focus to the other focus'. Reading III.48 as reflection is Heath's interpretation, not Apollonius' wording. Grade: Secondary. Read 2026-10-08.; MacTutor, Diocles, mathshistory.st-andrews.ac.uk/Biographies/Diocles/: Diocles 'was the first to prove the focal property of a parabolic mirror'; born about 240 BC, died about 180 BC; Toomer published the Arabic translation of On Burning Mirrors, found in the Shrine Library at Mashhad, in 1976. Heath 1921, vol. II p. 200, put Diocles 'a century or more later than Apollonius'. Grade: Secondary; Toomer's edition was not read. Read 2026-10-08.; Heath 1921, vol. II pp. 201 to 203, on the Fragmentum mathematicum Bobiense: the ray reflects to S with AS 'equal to ¼AL, where AL is the parameter' (Paraphrase); such burning mirrors 'bring about ignition at the point indicated' (the fragment, Attested in Heath's translation); 'an original which was probably not later than Apollonius' (Secondary). Read 2026-10-08.; Kepler, Ad Vitellionem paralipomena (Frankfurt, 1604), chapter IV, section 4, p. 93, archive.org/details/advitellionempar00kepl (page image of leaf 113 read), with the Latin as printed in C. Frisch, Joannis Kepleri astronomi opera omnia, vol. 2 (1859), p. 186, archive.org/details/joanniskeplerias02kepl: 'quae definitionem certam habent, nomen nullum'; the naming sentence as quoted; the parabola's second focus 'infinito intervallo a priore remotus', the 'caeco foco'; 1604 ed. p. 92 (leaf 112, end of IV.3), Frisch II p. 185: 'Memor autem eram eorum, quae Vitellio de speculo parabolico adurente scripserat'; 1604 ed. p. 96 (leaf 116, end of IV.4), Frisch II p. 188: 'ad ignes incendendos, ad infinite comburendum, consideratio earum plane est necessaria'. Grade: Attested; the English renderings are made here (W. H. Donahue's published translation was not consulted). Read 2026-10-08.; Lewis and Short, A Latin Dictionary, s.v. focus, at Perseus (perseus.tufts.edu): 'A fire-place, hearth'. Read 2026-10-08.; Frisch, Opera omnia vol. 2, p. 43: Kepler to Brengger, 'vere focus, eo quod illuc confluant omnes parallelae axi ex tota cavitate speculi; atque hic demum est violentissima incensio'; the reply is dated 'Pragae 17. Jan. 1605' (p. 53). Grade: Attested; Frisch prints excerpts, and the English rendering is made here. Read 2026-10-08.; G. G. Tailly, 'Extracorporeal shock wave lithotripsy today', Indian Journal of Urology 29(3), 2013, pp. 200 to 207, pmc.ncbi.nlm.nih.gov/articles/PMC3783700/: in an electrohydraulic source 'the shockwaves are generated in the first focus of a semiellipsoid reflector which then focuses the shockwaves into the second focus'. M. J. Semins and B. R. Matlaga, Indian Journal of Urology 26(3), 2010, pmc.ncbi.nlm.nih.gov/articles/PMC2978446/: electromagnetic sources are focused 'with either an acoustic lens, a parabolic reflector or ... at initiation'. Read 2026-10-08.; Wikipedia, Parabolic reflector (rev. 1368519182): parabolic mirrors 'gather light in reflecting telescopes and solar furnaces, and project a beam of light in flashlights, searchlights, stage spotlights, and car headlights'. Wikipedia, Cassegrain reflector (rev. 1358172906): 'A convex hyperbolic reflector has two foci and will reflect all light rays directed at one of its two foci towards its other focus.' NASA Science, Hubble Optics, science.nasa.gov/mission/hubble/observatory/design/optics/: 'Hubble's mirrors are curved hyperbolically'. Read 2026-10-08.; MacTutor, Germinal Pierre Dandelin, mathshistory.st-andrews.ac.uk/Biographies/Dandelin/: his 1822 memoir shows 'that if a cone is intersected by a plane in a conic, then the foci of the conic are the points where this plane is touched by the spheres inscribed in the cone'. The memoir itself was not read. Read 2026-10-08.; Computed here with Node on 2026-10-08: PF₁, PF₂ and the angles at the points in the try prompts, from sin ψ = b ÷ √(PF₁ × PF₂) for the ellipse (b = 3) and hyperbola (b = 4) and tan ψ = PL ÷ (2 × QV) for the parabola, and one-fourth of each figure (10 × 3.6 ÷ 4 = 9; 6 × (32 ÷ 3) ÷ 4 = 16, the hyperbola's latus rectum 32 ÷ 3 being about 10.67).

Check yourself · 4 cards · spaced review in the study dashboard

  1. recall

    Apollonius has no word for a focus. How does Conics Book III fix the two points, and for which curves?

    Show answer

    By another application of areas, for the ellipse and the hyperbola: each point lies on the axis where the rectangle formed by its distances to the two vertices equals one-fourth of the figure. For the ellipse the points lie between the vertices, for the hyperbola within each branch. He calls them 'the points arising out of the application'. Heath notes that the focus of a parabola is not used or mentioned by Apollonius.

  2. apply

    The ellipse in the foci figure has its axis 10 long, latus rectum 3.6 and foci 4 from the centre. With P at the right-hand end of the axis, what are PF₁, PF₂ and their sum, and how does this check Apollonius' rule for fixing the foci?

    Show answer

    PF₁ = 4 + 5 = 9 and PF₂ = 5 − 4 = 1, so the sum is 10, the axis, as it is at every point. F₂ is 1 from one vertex and 9 from the other, and 1 × 9 = 9, one-fourth of the figure 10 × 3.6 = 36.

  3. explain

    What does Conics III.48 prove, and whose reading turns it into a law of reflection?

    Show answer

    It proves that the lines from the two points to any point of the curve make equal angles with the tangent there. Read as a mirror, that means that in the ellipse a ray from one point reflects towards the other, and in the hyperbola away from it, as if the other had sent it. The mirror reading is Heath's, who thought it 'certain that Apollonius was aware' of it for the ellipse; Apollonius states only the equal angles.

  4. recall

    Who first called these points foci, in which work and year, and for what stated reason?

    Show answer

    Kepler, in Ad Vitellionem paralipomena (1604). The points had a definite definition but no name, and he called them foci 'for the sake of light and with our eyes on mechanics'. Focus is Latin for a hearth, but that is not the reason he gives.

VI · Eccentricity

One number sorts them all.

The directrix does not appear in the Conics at all. Heath finds the focus-and-directrix property in Pappus, who compiled his Collection in the fourth century AD, as a lemma to Euclid's lost Surface-Loci: a point whose distance from a given point is in a fixed ratio to its distance from a given line lies on a conic section. In Hultsch's Latin, Pappus says the curve is 'si proportio sit magnitudinis aequalis ad aequalem, parabolam, sin minoris ad maiorem, ellipsim, sin maioris ad minorem, hyperbolam': a parabola if the ratio is of equal to equal, an ellipse if of less to greater, a hyperbola if of greater to less. Because Pappus gives it as a lemma to Euclid, Heath infers that Euclid assumed it without proof. Heath (1921) called the lemma 'the first statement on record of the focus-directrix property of the three conic sections'. He wrote before Diocles' Arabic text came to light, and MacTutor, following Toomer, says Diocles had given a focus-directrix construction of the parabola. In the figure the fixed ratio is the eccentricity, e.

The figure fixes the focus F and a vertical directrix to its right, and keeps the nearest point of the curve 1 unit from F. D is the foot of the perpendicular from P to the directrix. The slider Eccentricity runs from 0 to 3.5, and the slider Point P moves P round F by its angle from the vertex, in degrees; you can also drag P. The readouts give the Section, PF, PD (to the directrix) and PF ÷ PD. The curve is drawn from PF = (1 + e) ÷ (1 + e cos θ), where θ is the angle at F from the vertex, with the directrix (1 + e) ÷ e from F, so PF ÷ PD equals e wherever P goes. Below 1 the curve closes, at 1 it opens into a parabola, and above 1 it is a hyperbola. The hyperbola's far branch lies beyond the directrix and keeps the same ratio to the same focus and directrix, and its asymptotes are drawn faint. As e falls to 0 the directrix moves off to infinity and the curve becomes a circle; strictly, a circle is not defined by a focus and directrix in the ordinary plane. The presets Circle (0), Earth (0.0167), Mercury (0.2056), Halley's Comet (0.9679), Parabola (1), 'Oumuamua (1.201) and Borisov (3.356) set real orbital eccentricities. 3I/ATLAS, at 6.141, lies beyond the slider.

Kepler had already set the curves in one order, speaking, in his words, 'analogice magis quam geometrice', more by analogy than geometrically: from the straight line through infinitely many hyperbolas to the parabola, then through infinitely many ellipses to the circle. He saw the foci follow the same order, with the parabola's second focus infinitely far away. Some modern writers call this Kepler's principle of continuity, which is a later name: the law of continuity is usually credited to Leibniz. Newton gave the family its physics. In the Principia (1687) he showed that a body moving in an ellipse, a hyperbola or a parabola about a focus is drawn towards that focus by a force inversely as the square of the distance (Book I, Propositions XI to XIII). His first corollary states the converse: under such a force a body 'will move in one of the conic sections, having its focus in the centre of force', the circle counting as an ellipse. Earth and Mercury have eccentricities well below 1, Halley's Comet comes close to 1, and 'Oumuamua and Borisov, above 1, follow hyperbolas. The Astronomy lab shows Kepler's law of equal areas on such an ellipse, and the Academy sets out his third law and Newton's derivation of it.

Try this

  • At the start (Eccentricity 0.500, Point P at 90°, straight above F), PF reads 1.50, PD (to the directrix) 3.00 and PF ÷ PD 0.500. Move P to 180°, the far end of the ellipse: PF reads 3.00 and PD 6.00, and the ratio is still 0.500.
  • Press Parabola (1): with P at 90°, PF and PD both read 2.00 and PF ÷ PD reads 1.000. Set the eccentricity to 1.5: PF reads 2.50, PD 1.67 and the ratio 1.500. The far branch appears beyond the directrix, and P can now swing less than 131.8° either side of the vertex before the near branch runs off along its asymptote.
  • Press Earth (0.0167), then Halley's Comet (0.9679). Earth's orbit is hard to tell from a circle. Halley's runs off the frame: its far end lies about 61 units from F, against 1 unit for the nearest point. Press 'Oumuamua (1.201), then Borisov (3.356), and the hyperbola opens much wider.

The nearest point is always 1 unit from F, so real orbits appear in shape, not in size: Halley's Comet came within 0.587 AU of the Sun at its 1986 perihelion (0.575 AU in its 1968 elements) and Borisov within 2.007 AU. Orbital eccentricities also drift as the planets pull on a body. JPL gives Halley's as 0.9679 in its 1968 elements and 0.9673 at its 1986 perihelion, and each preset is the value at one date.

Sources: Heath 1896, p. xxxvi: Pappus's second lemma to the Surface-Loci (Hultsch p. 1006 seqq.), 'the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed line is a conic section'; pp. xxxviii to xxxix: 'this property does not appear at all in Apollonius, and the focus of a parabola is not even mentioned by him'. Heath 1921, vol. II p. 119: 'It is remarkable that the directrix does not appear at all in Apollonius's great treatise on conics'; Pappus gives the proposition 'as a lemma to Euclid's Surface-Loci, from which we cannot but infer that it was assumed in that treatise without proof' (Secondary); p. 426: 'the first statement on record of the focus-directrix property of the three conic sections'. Read 2026-10-08.; Pappus, Collection VII, Prop. 238, in F. Hultsch's edition and Latin translation, vol. II (Berlin, 1877), archive.org/details/pappialexandrin01hultgoog. Hultsch prints the theorem twice: first as lemma II on the Surface-Loci (pp. 1005 to 1007, Heath's 'second lemma'), then repeated as lemma V (pp. 1013 and 1015), whose Latin is quoted here. Grade: Attested, in Hultsch's Latin; the Greek was not checked, and the English rendering is made here. Read 2026-10-08.; MacTutor, Pappus, mathshistory.st-andrews.ac.uk/Biographies/Pappus/: Pappus observed the solar eclipse at Alexandria of 18 October 320, and the Collection is thought to date from around 340. MacTutor, Diocles: 'Propositions 4 and 5 giving the focus directrix construction of the parabola' (after Toomer, whose edition was not read). MacTutor, Diocles, also: 'No writing of Diocles was known to Heath in 1921'. Read 2026-10-08.; Kepler, Ad Vitellionem paralipomena (1604), chapter IV, section 4, p. 92 (leaf 112, page image read) and pp. 94 to 95 (leaves 114 to 115); Frisch, Opera omnia vol. 2, pp. 185 to 187: 'Inter has lineas hic est ordo causa proprietatis suae, et analogice magis quam geometrice loquendo: quod a linea recta per hyperbolas infinitas in parabolen, inde per ellipses infinitas in circulum est transitus'; 'In media parabole infinito intervallo distant'. Grade: Attested; the English renderings are made here. Read 2026-10-08.; Wikipedia, Conic section (rev. 1377246756): Kepler 'extended the theory of conics through the "principle of continuity"'; 'A circle is a limiting case and is not defined by a focus and directrix in the Euclidean plane.' Wikipedia, Law of continuity (rev. 1297183868): 'a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler'. Grade: Secondary. Read 2026-10-08.; Newton, The Mathematical Principles of Natural Philosophy, translated by Andrew Motte (London, 1729), vol. 1, archive.org/details/bim_eighteenth-century_principia-english-th_newton-sir-isaac_1729_1, and Wikisource: Book I, Propositions XI (pp. 79 to 80), XII (pp. 81 to 82) and XIII (pp. 84 to 85), each finding the centripetal force towards the focus 'reciprocally in the duplicate ratio of the distance'; Corollary 1 to Proposition XIII, p. 85, as quoted; 'In these corollaries, I consider the circle as an ellipsis' (p. 86). Grade: Attested, in Motte's translation. Read 2026-10-08.; NASA/JPL Small-Body Database API, ssd-api.jpl.nasa.gov/sbdb.api (full-prec=true): Halley's Comet e 0.9679, perihelion distance 0.575 AU (osculating, epoch 1968-01-20, solution 75); 'Oumuamua e 1.201, 0.256 AU (epoch 2017-11-23); Borisov e 3.356, 2.007 AU (epoch 2020-01-05); 3I/ATLAS e 6.141, 1.356 AU (epoch 2026-02-19, in a solution that models nongravitational accelerations). JPL Horizons API, Halley's osculating elements at 1986-02-09: e 0.9673, perihelion distance 8.783 × 10⁷ km = 0.587 AU. NASA, Earth Fact Sheet and Mercury Fact Sheet, nssdc.gsfc.nasa.gov/planetary/factsheet/: orbit eccentricity 0.0167 and 0.2056. Read 2026-10-08.; Computed here with Node on 2026-10-08 from PF = (1 + e) ÷ (1 + e cos θ) and PD = (1 + e) ÷ e − PF cos θ: the readouts in the try prompts, the near-branch limit of 131.81° for e = 1.5 (where cos θ = −1 ÷ e), Halley's far end at (1 + e) ÷ (1 − e) = 61.3 units, and the turn of the path of 3I/ATLAS, 2 × arcsin(1 ÷ e) = 18.7° (about 19°) for e = 6.141.

Check yourself · 4 cards · spaced review in the study dashboard

  1. recall

    The directrix does not appear in Apollonius' Conics. Where did Heath find the focus-and-directrix property, and how does the ratio sort the curves?

    Show answer

    In Pappus' Collection (fourth century AD), as a lemma to Euclid's lost Surface-Loci: a point whose distance from a given point is in a fixed ratio to its distance from a given line lies on a conic section. The curve is a parabola if the ratio is of equal to equal, an ellipse if of less to greater, and a hyperbola if of greater to less.

  2. apply

    The eccentricity figure keeps the nearest point 1 unit from the focus F, so PF = (1 + e) ÷ (1 + e cos θ) and the directrix lies (1 + e) ÷ e from F. With e = 0.5, what are PF, PD and PF ÷ PD when P is at 90° and at 180°?

    Show answer

    At 90°, cos θ = 0, so PF = 1.5. The directrix is 1.5 ÷ 0.5 = 3 from F, and P is straight above F, so PD = 3 and PF ÷ PD = 0.5. At 180°, cos θ = −1, so PF = 1.5 ÷ 0.5 = 3; P lies 3 on the far side of F, so PD = 3 + 3 = 6, and the ratio is still 0.5.

  3. recall

    In what order did Kepler set the conic sections, and how did he qualify the claim?

    Show answer

    From the straight line through infinitely many hyperbolas to the parabola, then through infinitely many ellipses to the circle, with the foci following the same order and the parabola's second focus infinitely far away. He said he was speaking 'analogice magis quam geometrice', more by analogy than geometrically. Calling it his 'principle of continuity' is a later name.

  4. connect

    What did Newton's Principia add to the family of conics, and how do the orbit presets in the eccentricity figure fit it?

    Show answer

    Its physics. A body moving in an ellipse, a hyperbola or a parabola about a focus is drawn towards that focus by a force inversely as the square of the distance (Book I, Propositions XI to XIII), and Corollary 1 gives the converse: under such a force a body moves in one of the conic sections, with its focus at the centre of force. Earth (0.0167) and Mercury (0.2056) are well below 1, Halley's Comet (0.9679) comes close to 1, and 'Oumuamua (1.201) and Borisov (3.356), above 1, follow hyperbolas.