Chap. VIII.

GEOMETRY.

761

Fig. 787.

may be also calculated ; this is an important preliminary before any constructions are com-menced, and great accuracy in the drawing is requisite to make the estimates correctly.

An orthographic plan shows the extent as well as eleva-tion of the several portions of the building.

A sctnographic plan represents the whole in perspective.

Vitruvius informs us that architecture depends on fit-ness and arrangement, and also on proportion, uniformity,consistency, and economy : the first relates to the niceadjustment of the dimensions of the several parts to thewhole, as well as to their use. Ordination is the wordwe have rendered into fitness, and in it is comprised theterms we are now using, as Ichnography, Orthography ,and Scenography ; as the first relates to the plan drawngeometrically, so does the second to the elevation ; thelast exhibits the front and receding side properly shadowed,the lines being drawn to their proper vanishing points.

After such a description as is contained in the authorabove cited, it is not possible to fancy that the Homansw’ere unacquainted with our method of making designs,and particularly when he further observes that the threesystems of preparing them are the result of thought andinvention, the first being an effort of the mind, ever in-cited by the pleasure attendant on success in compassingan object, whilst the other is the effect of this effort, whichshows a new light on things the most recondite, and pro-duces them to answer the intended purpose : such are theends of arrangement. Proportion is that agreeable har-mony which results from one and all parts agreeingwith each other and the whole : uniformity is the parityor likeness of the parts to one another : consistency isthe result found when the work exhibits a suitable detail,and economy is the due and proper application of themeans afforded, and prudently employing it. The secondchapter of the first book of Vitruvius should be studiedby all who are desirous of becoming civil engineers.

Of different sites or situations of planes with regard tothe horizon , which is said to be either sensible or rational;tlie first is a plane which is a tangent to the earth’s surfaceat the place of the spectator, extended on all sides till itis bounded by the sky ; the latter is a plane, parallel tothe former, but passing through the centre of the earth.

These two terms are only relative, as they vary with thespectator’s position; for when his eye is in the plane of thesensible horizon he can only see what is above it, but whenit is raised above the horizon he can observe what is beneathit. The sensible horizon is therefore properly defined tobe the conical surface which has its apex in the eye ofthe spectator, and embraces the portion of the earth overwhich the eye can reach ; the visual rays, which are tan-gents to the earth, are situated in this surface, and pointbelow' the true sensible horizon, or the rational horizon

-^T-

- A

_— |

—-1$-

j-—! c

— -

—I

Fig. 788.

Fig. 789.

which is parallel to it; and the angle which a visual ray makes with the plane of thehorizon is termed its dip or depression , and which is easily estimated from the known dimen-sions of the earth, and the height of the eye above its surface. An inclined plane is thatwhich is neither horizontal nor vertical, but which slopes on the horizon, as the clifT at D.

A may be considered the true level or the surface of the waters ; B is the horizontal planeparallel with A; C is a vertical plane perpendicular to A, or parallel to the plummet let fallat E; D may be called an inclined plane.

Of Sines, Tangents , and Secants _The sine of any arc of a circle is the straight line drawn

from one extremity of the arc perpendicular to the radius, passing through the other ex-tremity. The sine of an arc is the half of the chord of the double arc. The tangent to acurve is a straight line which meets or touches the curve without intersecting it; thearc and its tangent have always a certain relation to each other, and when the one isgiven in parts of the radius, the other can always be computed by means of an infiniteseries; the Arabians were the first to introduce tangents into trigonometry, and whichrender important service in simplifying many calculations. The secant is a straight linedrawn from the centre of a circle to one extremity of an arc, and produced until it meetsthe tangent to the other extremity. The secant of an arc is a third proportional to the