The first assertion is practically that of tangency,--"will fall outside the circle." The second one states, substantially, that there is only one such tangent, or, as we say in modern mathematics, the tangent is unique. The third statement relates to the angle formed by the diameter and the circumference,--a mixed angle, as Proclus called it, and a kind of angle no longer used in elementary geometry. The fourth statement practically asserts that the angle between the tangent and circumference is less than any assignable quantity. This gives rise to a difficulty that seems to have puzzled many of Euclid's commentators, and that will interest a pupil: As the circle diminishes this angle apparently increases, while as the circle increases the angle decreases, and yet the angle is always stated to be zero. Vieta (1540-1603), who did much to improve the science of algebra, attempted to explain away the difficulty by adopting a notion of circle that was prevalent in his time. He said that a circle was a polygon of an infinite number of sides (which it cannot be, by definition), and that, a tangent simply coincided with one of the sides, and therefore made no angle with it; and this view was also held by Galileo (1564-1642), the great physicist and mathematician who first stated the law of the pendulum. THEOREM. _Parallel lines intercept equal arcs on a circle._ The converse of this proposition has an interesting application in outdoor work. [Illustration] Suppose we wish to run a line through _P_ parallel to a given line _AB_. With any convenient point _O_ as a center, and _OP_ as a radius, describe a circle cutting _AB_ in _X_ and _Y_. Draw _PX_. Then with _Y_ as a center and _PX_ as a radius draw an arc cutting the circle in _Q_. Then run the line from _P_ to _Q_. _PQ_ is parallel to _AB_ by the converse of the above theorem, which is easily shown to be true for this figure. THEOREM. _If two circles are tangent to each other, the line of centers passes through the point of contact._ There are many illustrations of this theorem in practical work, as in the case of cogwheels. An interesting application to engineering is seen in the case of two parallel streets or lines of track which are to be connected by a "reversed curve." [Illustration] If the lines are _AB_ and _CD_, and the connection is to be made, as shown, from _B_ to _C_, we may proceed as follows: Draw _BC_ and bisect it at _M_. Erect _PO_, the perpendicular bisector of _BM_; and _BO_, perpendicular to _AB_. Then _O_ is one center of curvature. In the same way fix _O'_. Then to check the work apply this theorem, _M_ being in the line of centers _OO'_. The curves may now be drawn, and they will be tangent to _AB_, to _CD_, and to each other.