Select a body and drag the planet around its epicycle. Watch the deviation rise, cross zero, go negative, return. The smooth, symmetric pattern is the structure Āryabhaṭa noticed in the residuals — and encoded as a 24-entry sine table.
jya(180°−θ) = jya(θ), jya(180°+θ) = −jya(θ), jya(360°−θ) = −jya(θ). Twenty-four numbers cover the entire circle. The correction becomes a lookup. See a worked example ↓
The gray dot on the dashed circle is the mean planet — it moves at perfectly uniform speed, easy to compute from time alone. The orange dot is the true planet, riding a smaller circle (the epicycle) centered on the mean. The green arrow shows the deviation — the jya — between the two. The dashed circle is the body's deferent — its orbit around the Earth in Āryabhaṭa's geocentric system.
One thing worth noticing: at θ = 0° and θ = 180°, the orange dot sits in line with the gray dot as seen from Earth — same longitude, just a different distance. The green arrow vanishes because what it measures is the perpendicular displacement, the part that actually changes the planet's angular position in the sky. The equation tracks longitude, not distance. The epicycle moves the planet both radially and laterally; only the lateral part shows up as a correction.
What makes the argument click: pause the animation and drag the epicycle slider alone. The true planet traces a loop around the mean position, and the deviation value oscillates — rising, crossing zero, going negative, returning. That smooth, symmetric pattern is exactly what Āryabhaṭa noticed in the residuals after subtracting uniform motion from observation. It wasn't noise; it was structure.
The equation in the panel formalizes it: true = mean ± k · jya(θ). The sine table encodes the jya values so that correction becomes a lookup, not a fresh geometric calculation. The table is a compressed map of deviation — and once you see it that way, it stops being a list of mysterious numbers.
The k value is a property of the body, not a free parameter. Each body has its own manda epicycle ratio that Āryabhaṭa derived from observation, ranging from k ≈ 0.038 for the Sun (whose motion is nearly uniform) to k ≈ 0.20 for Mars. Switch bodies in the panel to see the same geometric model applied with each body's historical ratio. The Moon's correction is moderate and clearly visible; the Sun's is barely perceptible; Mars produces the largest loop.
The deferent radius also varies between bodies, reflecting the geocentric ordering used in Indian astronomy: Moon (innermost) — Mercury — Venus — Sun — Mars — Jupiter — Saturn (outermost). This ordering was derived from orbital periods (slower-moving bodies were assumed to be farther). The proportions shown here aren't astronomically literal — true geocentric distances would put Saturn around thirty times farther than the Moon, which would push inner bodies past the limit of usable resolution — but the ordering is correct, and so is the qualitative observation that Mars and Saturn occupy larger orbits while the Moon hugs the Earth.
For the five planets (Mercury, Venus, Mars, Jupiter, Saturn), the visualization shows only the manda (slow, equation-of-center) correction. These bodies also require a śīghra correction — a second, larger epicycle that accounts for the apparent retrograde motion produced by Earth's own orbital motion. The śīghra is not shown here. The Moon and Sun need only manda, and the model you see is complete for them.
Āryabhaṭa's sine table has 24 entries at intervals of 3°45' (3.75°) from 3.75° to 90°. The values are given in the Āryabhaṭīya, Gaṇita 12, expressed in arcminutes on a base radius of R = 3438' — a value carefully chosen so that at small angles, the sine in arcminutes is numerically close to the angle itself in arcminutes (3438' ≈ one radian expressed in arcminutes).
A subtle but important detail of the original tabulation: Āryabhaṭa did not list the cumulative sine values directly. He listed the differences between consecutive entries — 225, 224, 222, 219, 215, 210, 205, 199, and so on down to 7 — and the cumulative sums give the sine values shown in the table above. This is what makes interpolation effortless. The "next difference" needed for linear interpolation is already there, indexed by position.
The table covers only the first quadrant (0° to 90°). Three symmetry rules extend it to the full 360° circle. The rules become obvious once you remember that sine is the y-coordinate of a point on a unit circle: because the circle is symmetric across both axes and through the origin, knowing the height of one point automatically gives you the height of three other points related to it by reflection.
jya(180° − θ) = jya(θ) — reflection across the y-axis. Same vertical height, mirrored horizontally.
jya(180° + θ) = −jya(θ) — reflection through the origin. Same magnitude, flipped to negative.
jya(360° − θ) = −jya(θ) — reflection across the x-axis. Same horizontal position, vertically flipped.
The table genuinely only needs 24 entries. The rest is a matter of recognizing the quadrant and applying a sign.
Suppose the planet's epicycle angle is θ = 175° (in Q2). What is jya(175°)? Here is the actual computational sequence a practitioner would follow:
jya(180° − θ) = jya(θ). So 180° − 175° = 5°. This is the angle to look up.A realistic Āryabhaṭa-era practitioner would do this in roughly fifteen seconds, mentally, without any instrument:
Q2 — flip to 5°. Bracket between 3.75° and 7.5°. The next difference is 224. One-third of 224 is about 75. So 225 + 75 = 300. Positive, because Q2.
The 24-entry table plus three symmetry rules turns trigonometry from geometric construction into a workflow — fast enough to compute dozens of planetary positions in an evening of work, with no apparatus more elaborate than a memorized table and an arithmetic head.
The sine function existed as a geometric object in many ancient traditions. The Greeks had chord tables — Hipparchus and Ptolemy worked with the chord of an angle (essentially 2·sin(θ/2)) and could compute it for any angle they needed by geometric construction. Islamic astronomers later refined the values to remarkable precision.
But Āryabhaṭa's table-plus-symmetry combination — tabulation in differences, indexed for fast lookup, structured around quadrant symmetry — is something else. It turns trigonometry from a thing you compute into a thing you consult. That is the difference between a science and a workflow.
This is the kind of move an enabling tradition produces. Not the discovery of the sine function — that was widely available across multiple civilizations. The discovery is the encoding: recognizing that a periodic function can be reduced to twenty-four numbers plus three rules, and that the right way to present those twenty-four numbers is as differences rather than as cumulative values. The encoding is what makes the technology deployable. The encoding is the breakthrough.