Compass and straightedge

Measure & Number

“Thou hast ordered all things in measure and number and weight.”Wisdom of Solomon 11:20

“When he prepared the heavens, I was there: when he set a compass upon the face of the depth.” Wisdom speaks so in Proverbs 8:27, and the painters of the thirteenth century took it at its word: in the great moralised Bibles God bends over the world with a pair of compasses, the architect of all that is.

The masons who built the cathedrals had the same two tools, a compass and a straightedge. With them they set out every arch, every rose and every light. Each figure here is drawn the way they drew it, from a stated centre and a stated opening, step by step. Nothing is traced. Where a meaning is a later reading, we say so.

ExactEvery point and every opening comes from the figure itself.ConstructedEvery point is found; a few openings are chosen by eye, as a mason chose a moulding.ApproximateNo exact construction exists, and the gap is measured.
Plate IExact

Vesica piscis

Two circles of one radius, each set on the centre of the other.

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Plate IIConstructed

Six petal rosette

One compass opening, never changed.

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Plate IIIConstructed

Trefoil

Three lobes centred on an equilateral triangle, one of the commonest openings in Gothic window tracery.

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Plate IVConstructed

Quatrefoil

Four lobes on the corners of a square.

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Plate VExact

Three arches on one span

Each flank is an arc swung from a pin on the springing line.

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Plate VIExact

Two lights and an oculus

An equilateral arch on the span, two smaller ones inside it, and a circle whose centre is found by one arc swung from the springing through the quarter point.

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Plate VIIConstructed

Rose of twelve

Walk the compass round the rim at its own opening and it closes in six steps.

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Plate VIIIExact

Triquetra

Three circles on the corners of an equilateral triangle, each passing through the other two centres, so every pair is a vesica.

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Plate IXConstructed

Five petal rose

Five lobes on the five steps of a true pentagon, which Euclid constructs exactly (Book IV, Proposition 11).

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Plate XExact

Pentagon, after Ptolemy

A diameter, its perpendicular, the midpoint of one radius, and one arc from that midpoint through the top of the circle.

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Plate XIApproximate

Pentagon, one opening

The compass is set once and never moved: the method of the workshop booklet Geometria deutsch, printed again by Dürer in 1525.

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Plate XIIConstructed

Shield of the Trinity

A medieval teaching diagram: three circles on a circle and one at its centre, joined by six bands.

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Plate XIIIConstructed

Chi rho

The first two letters of Christ’s name in Greek, chi and rho, set one through the other.

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Plate XIVConstructed

The four evangelists

A quatrefoil set on the cross, each lobe lettered with an evangelist’s name in Latin and his living creature in the words of the Vulgate.

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Plate XVExact

Jerusalem cross

A cross potent, its arms ending in crutches, between four small crosses.

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Plate XVIConstructed

Heater shield, gules

Here the flanks are arcs whose radius is the full width of the shield, each swung from the foot of the opposite side, so the point is the apex of a vesica.

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Plate XVIIConstructed

Spanish shield, azure

Straight sides closed by one half circle about their foot.

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Plate XVIIIApproximate

Roriczer’s shield

“So einer ein schilt mit der geometry machen wil”: the workshop booklet Geometria deutsch, attributed to Mathes Roriczer, builds a shield from a square under the chief, a diagonal swung down the axis, and one arc for the base about a point between.

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Plate XIXApproximate

Seven will not close

No compass and straightedge can draw a true seven sided figure: Gauss stated it in 1801 and Wantzel proved it in 1837.

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Plate XXApproximate

Nine from six

Dürer’s nonagon of 1525.

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Plate XXIFreely set

House mark

A stem with crossbars, struts and chevrons, each named by its place.

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