Thomas Goldstein
Dawn of Modern Science

In explaining various types of projection, Ptolemy had touched on the problem of foreshortening and suggested a method for its cartographic representation. Alberti realized the underlying problem for artists was basically the same. He took a clue from Ptolemy and applied his mathematical concept to the type of foreshortening involved in the simple visual perception that is represented in a painting. Generations of Renaissance painters and bas-relief sculptors from then on could indulge in the accurate depicting of depth, the problem with which artists had wrestled since the time of Giotto, introducing a major dimension of realism and drama—all thanks to a cartographic concept.

Map projection and perspective painting, in other words, were developing plong parallel lines in a mutually fruitful relationship for some hundred and fifty years, or across most of the Renaissance. In fact, the method for projecting the curved surface of the earth on a map while keeping actual distances in their proper proportion, and the painter's technique in capturing perspective images on a canvas in the exact geometrical order in which they appear to the eye are two ways of dealing with the same problem. Whether one wishes to represent a large portion of the earth, or merely the tiny sector the eye comprehends at a glance, the mind is, in principle, faced with the identical challenge. In both cases a window is opened, so to speak, on a segment of earthly space, whether small or large—just as Giotto had 'opened a window' on the world when he broke through those bland backgrounds of the traditional Medieval paintings—and the three-dimensional image presenting itself to us is then reduced to flat shapes on a two-dimensional surface, in a way that suggests the true distances and proportions. (Alberti in his book on perspective actually suggests that the problem facing the painter may be reduced to the appearance of three-dimensional objects on a windowpane.)

  The World was all before them, where to choose
Their place of rest, and Providence their guide:
They, hand in hand, with wand'ring steps and slow,

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Through Eden took their solitary way.