This section is from the book "Modern Buildings, Their Planning, Construction And Equipment Vol5", by G. A. T. Middleton. Also available from Amazon: Modern Buildings.
(Contributed by Walter Hooker)
In considering the application of the principles of masonry to practical use the object has been to eliminate from the examples given, as far as possible, the introduction of calculations of an abstruse nature. By this means it is hoped to bring within the understanding of both the mason and the student such principles as are needful in meeting with and solving all ordinary problems that may occur in actual practice.
As a considerable amount of skill both in plane and solid geometry, and also of projection, is required to enable the mason to determine the shapes of the stones of the various features of a building, this chapter is devoted to the subjects above named; and the several diagrams and their explanations which it contains will be found sufficient to clearly and concisely demonstrate the more useful problems that may be required in ordinary building work. One or more problems are inserted that more strictly apply to engineering, but their utility will perhaps be appreciated as showing the principles of stereotomy outside of the architect's legitimate sphere.
Each problem is illustrated by outline diagrams, and in the more advanced stages by perspective views or by isometric drawings.
It is presumed that the reader has already a fair acquaintance with the rudiments of practical geometry, algebra, and the elements of Euclid. It would be impossible otherwise within the limits of this work to give sufficiently detailed instruction to enable the student to benefit materially by its introduction.
To divide a line in extreme and mean ratio.
Let AB (Fig. 76) be the line. On A erect the perpendicular AC = 1/2 AB. Join CB. With C as centre and radius CA draw an arc cutting CB in D. With centre B and radius BD draw an arc cutting AB in E. AB is then divided in extreme and mean ratio at E.
That is, Ae Eb: Eb Ab.
To find a mean proportional between two given lines.
Let AB be the greater and C the lesser line (Fig.77). Prolong AB indefinitely to X, and cut off BD = C. With AD as diameter draw a semicircle Aed. Draw BF at right angles to AD to meet the semicircle in F. Then FB is a mean between AB and BD, which is =C.
That is, Ab Bf Bf Bd or C.
To find a fourth proportional to three given lines
Let A, B, and C be the three given lines (Fig. 78). At any convenient angle with one another, say 35 or 40 degrees, draw DE and DF. From DE cut off