non-existent numbers of my arithmetical brother, nor my immaterial and imaginary lines. Give me leave to repeat, in spite of your indignation, that though they are (in this present visible world of ours) non-existent, yet these lines and numbers are ‘realities.’ That they are realities, and that our conclusions about them are real and true, is proved by the one test of truth: our conclusions _work_. Our discoveries are in harmony with the universe. A perfect circle you never saw and never will see: yet it is as real as a beefsteak and a pint of porter. I believe in a perfect circle by Faith; I accept it with reverence as an impression, if I may so dare to speak, on the Mind of the Universe, which He has communicated to me. What is more, I believe that He intended us to study this and other immaterial realities that our minds might approximate to His. Take a cone, my practical friends. What do you see in it? Nothing, I fear, except a shape that reminds you of an extinguisher or a fool’s cap. Yet this little solid contains within itself the suggestions of all the mysteries of motion in heaven and earth. Slice your cone parallel to the base: there you have the perfect circle. Slice it again, parallel to one of the sides: there you have the parabola, the curve of terrestrial motion. Slice it once more, midway between these two sections: there you have the ellipse, the curve of celestial motion for which all the astronomers were seeking in vain through something like a score of centuries. Seriously now, my half-educated friends, in spite of the sense you may for the most part entertain of your own importance, do you not in your more modest moods sometimes feel inclined to say that, ‘A circle is, after all, a reality, perhaps more real than I am myself’?” What do you think of all this? For my part, I am inclined to think the Mathematician has the best of it. A good deal will turn upon the meaning of that dangerous word “reality,” about which I will give you my notions, perhaps, hereafter.[2] But even if you dispute his assertions about the reality of his “ideas,” you cannot, I am sure, deny the immense practical importance, as well as the universal acceptance, of his conclusions and discoveries; and you will do well to remember that this immensely important, this undisputed and indisputable knowledge, could never have been attained if we had not called in the Imagination to create for us ideas that never will be, and never can be, realised in this present material world. Let us pass now from knowledge about things to knowledge about persons, _i.e._ about actions and motives. Our knowledge about actions depends on (1) personal observation; (2) testimony; (3) circumstantial evidence or any combination of these three.