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.
Scientia Perspectiva
Overview 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.
Punctorum Relatio ad Planum Spectatoris
Overview 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.
Origo Regularum Artis Perspectivae
Overview of Origo Regularum Artis Perspectivae (1680 – 1681)
In Origo Regularum Artis Perspecitvae quales sine libro ac Magistro inveni (LH 35 XI 17 Bl. 19-20), Leibniz analyses the case of the perspective representation of a segment floating above the objective plane. Just like in other manuscripts about perspective, similar triangles are used to establish relations between different line segments, but this time the terminology used by Leibniz in similar writings, for example the use of “inclinatio” and “declinatio” in Punctorum relatio to indicate how far a given point stands with respect to an established principal point, is substituted here with a terminology taken from geography (“imitatione Geographorum”): the distance between the eye and the Tabula will be called “longitude of the eye”, and with respect to an objective point, the distance to the left or to the right from the principal point will be called “latitude of the object”, while its distance from the tabula will be called “longitude of the object”. In the same fashion, every length which indicates how high an object is with respect to the eye of the spectator will be called “altitude”. We find here one of the most mathematically developed expression of the theorems already present in the other manuscripts, since a proper formula for the general rule of perspective is given. The watermark indicates that the manuscript was written after 1680 – 1681, but no other information can be deduced.
Trigonometria Sphaerica Tractanda per Projectionem in Plano
Overview 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.
Auxilia Calculi ex Ductu Linearum
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.