Manuscripts on Perspective and Projective Geometry
This section is devoted to Leibniz's work on geometrical perspective and its connection with the studies on the projection of curved geometrical objects. Leibniz already piqued an interest in perspective during his sojourn in Paris in the early years, but the development of an original theory on the topic began around 1680 and it had no significant developments after 1687. Leibniz's aim was that of simplifying, with his new method, the relationship between the observer and the geometrical objects belonging to the objective and the apparent planes. Facing the challenge of establishing these relationships in the case of a curved apparent plane, he began to study in the same years the case of the projection of a sphere on a plane.
Overviews
Overview of Auxilia Calculi ex Ductu Linearum (1680 – 1716)
This text doesn’t concern directly perspective, but it shows a very important rule used in it, as Leibniz himself writes at the end of the manuscript. It deals with the properties of similar right triangles, one of which is generated by drawing a segment between a cathetus and the hypotenuse, perpendicular to this cathetus. The common vertex of the two triangles will be, in its application in perspective, the eye of the spectator, as it will be clear in the first part of Scientia perspectiva.
viewOverview of Origo Regularum Artis Perspectivae (1680 – 1681)
viewOverview of Punctorum Relatio ad Planum Spectatoris (1684 – 1687)
This manuscript probably belongs to a period between Leibniz’s beginnings on perspective and his last attempts, as it shows a similar approach to the later writings, yet less refined. Here we find one of Leibniz’s most important theorems about perspective: the ratio between the two coordinates used to identify a point in the apparent and in the objective plane (“inclinatio” and “declinatio”) is equal to the ratio between the distance of the spectator both from the tabula and the objective plane. The difference with the later writings is that here an unnecessary third plane is introduced, that is the plane parallel to the tabula, in which belongs the eye of the spectator. Since this introduction leads to a slightly different formulation of the theorem and considering Leibniz's aim for simplicity in perspective's writings, there are reasons to believe that Punctorum Relatio ad Planum Spectatoris precedes Scientia perspectiva.
viewOverview of Scientia Perspectiva (1684 – 1687)
Dated between 1684 and 1687, Scientia perspectiva may very well be Leibniz's last attempt to develop a comprehensive theory on geometrical perspective. Ideally, the manuscript could be divided in five parts: a first introductory part on the concepts involved in perspective and the main aim of this science, a second part concerning a general example of Leibniz's method and its fundamental theorem, a third part on how geometrical objects appear in the apparent plane with respect to the objective plane, a fourth part with a more sophisticated example of how similar triangles are involved in determining the various points in the apparent plane and a fifth, last part where Leibniz develops a general rule for perspective, concerning parallel lines, converging lines and relative points.
viewOverview of Trigonometria Sphaerica Tractanda per Projectionem in Plano (1679 – 1680)
This manuscript doesn’t concerns directly perspective, but it deals with the projection of a part of a sphere on a plane. However, by doing so Leibniz utilizes a similar method to that of perspective and we believe it could be the key to understand how curves and curved planes are represented in perspective, something that Leibniz always says it is possible to achieve but that he never shows in practice in the manuscripts.
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