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Paris Manuscript A — 89

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Bibliothèque de l'Institut de France, Paris

A series of geometrical figures, for considerations involving perspective and foreshortening. A square.

A square whose top and bottom are extended by parallel lines. An isosceles triangle is inscribed within the square, its vertex labeled n. The square and triangle are divided by a horizontal line somewhat lower than the middle of the height of the triangle. A visual line extends upward from one of the lower two vertices of the square or triangle, crossing the upper parallel line at point m. Labels:

m n

A square whose top and bottom are extended by parallel lines. A triangle shares the square's base, labeled S and t, and its vertex, labeled x, is on the upper parallel, exterior to the square, giving the impression that the triangle has tilted forward from the square. A horizontal line, labeled n and m, divides the triangle, as in the preceding drawing; the square shows position of the same horizontal divider, prior to the "tilting." A visual line extends from vertex t upward to meet the upper parallel at point v. Labels:

v x – n m - S t

A figure analogous to figure 2, inverted. An isosceles triangle is inscribed within the square, its base corresponding to the top of the square, labeled c and f, its lower vertex labeled n. The square and triangle are divided by a horizontal line somewhat above the middle of the height of the triangle, intersecting with the triangle at points r and t. A visual line extends downward from vertex f of the square, meeting the lower parallel at point o. Labels:

c f - r t - o n

A square, divided in half vertically, with a horizontal divider somewhat below the top of the square. An isosceles triangle shares the square's base; its apex coincides with the intersection of the vertical and horizontal dividers of the square.

A square whose top and bottom are extended by parallel lines. Two isosceles triangles share the square's base; the apex of the larger coincides with the top of the square, that of the smaller, with a horizontal divider of the square, somewhat below the middle of the height of the square. A visual line extends upward across the diagram from the lower far vertex of the square, to meet the upper parallel.

A square

Square with inscribed triangle, sight-line to m

m n

Triangle tilted forward, vertex x

v x – n m – S t

Inverted triangle, divider r t, sight-line to o

c f – r t – o n

Square with triangle apex at the dividers' crossing

Square with two nested triangles and a sight-line

Geometric Figures for Perspective and Foreshortening

Squares with inscribed triangles tilted forward to study foreshortened reductions

Geometry Painting & Perspective

The folio is a set of geometric constructions studying how upright figures foreshorten when tilted toward the ground. Squares carry inscribed isosceles triangles and horizontal dividers; in several the triangle is shown tilted forward or inverted, with sight-lines drawn from base vertices to points such as m, v and o on an upper parallel. The sheet is almost wordless, only diagram letters (m n; v x, n m, S t; c f, r t, o n) accompanying the drawings, with faint pencil repetitions of the same figures at the right.

On this page

Square with inscribed isosceles triangle, vertex n

A square, its top and bottom extended by parallels, holds an inscribed isosceles triangle with vertex n and a horizontal divider below mid-height. A sight-line rises from a lower vertex to cross the upper parallel at m, marking how the divided figure reads once projected.

The triangle tilted forward from the square (vertex x)

The same figure is shown tilted forward: the triangle shares the square's base S t but its vertex x lands on the upper parallel outside the square, as though it had fallen forward. The divider n m and a sight-line from t up to v record the foreshortened position against the upright figure.

Reader's Edition — described with AI from the e-Leo transcription.