by Gregory St. Vincent

Books I & II translated and annotated by
Ian Bruce

This is the start of a truly mammoth book running to some 1250 pages. At present only Books I and II have been translated here. The work received a lukewarm reception at the time (1647) as Gregorius asserted that he could square the circle, as the title indicates. However, there is a place for this work in the history of mathematics, as it was one of the forerunners of the theory of integration, and the natural logarithm was developed from geometic progressions applied to hyperbolic segments - though the present work does not extend this far. Finally a pdf version of the original work is being used in the translation, which gets rid of the problems associated with microfiche.
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The First Book establishes the geometrical tools to be used; in particular the extended use of proportional quantities. Thus, tasks we now accomplish using algebra are to be performed using classical Greek geometry instead.
Book I, Part I : Proportions between line segments.
Book I, Part II
: A large number of theorems on properties of triangles and their uses.
Book I, Part III : Concerning powers of line segments.

Book II is concerned with Geometrical Progressions, considered geometrically.
Book II, Part I : There is a long introduction on the use of geometric progressions and the limiting process; for Gregorius had a fascination for Zeno's Paradox. The starting conditions for a G.P. are considered.
Book II, Part II : A large number of geometric progressions are summed to infinity geometically to produce a finite limit. I wish to acknowledge the contributions made by Bob Burn (University of Exeter) regarding Book II, Part II, and I have been pleased to implement the changes he has suggested. In addition, [Feb. '09] he has suggested stricter translations in the statement of Th. 116, and in the following corollary, and these have been amended.
Book II, Part III : Suitable geometric progressions are summed to infinity geometically to produce the areas of planar surfaces.
Book II, Part IV : Suitable geometric progressions of cubes are summed to infinity geometically to produce the volumes and surface areas of certain cubical pyramids which are compared with parallelipeds.

Ian Bruce. Oct. 2007 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for personal or educational use. If you have any useful information regarding this translation, I would appreciate hearing from you. IB.