and may be called *simple identities*. I may instance, in the first place, those most elementary propositions which express the exact similarity of a quality encountered in two or more objects. I may compare the colour of the Pacific Ocean with that of the Atlantic, and declare them identical. I may assert that “the smell of a rotten egg is like that of hydrogen sulphide;” “the taste of silver hyposulphite is like that of cane sugar;” “the sound of an earthquake resembles that of distant artillery.” Such are propositions stating, accurately or otherwise, the identity of simple physical sensations. Judgments of this kind are necessarily pre-supposed in more complex judgments. If I declare that “this coin is made of gold,” I must base the judgment upon the exact likeness of the substance in several qualities to other pieces of substance which are undoubtedly gold. I must make judgments of the colour, the specific gravity, the hardness, and of other mechanical and chemical properties; each of these judgments is expressed in an elementary proposition, “the colour of this coin is the colour of gold,” and so on. Even when we establish the identity of a thing with itself under a different name or aspect, it is by distinct judgments concerning single circumstances. To prove that the Homeric χαλκός is copper we must show the identity of each quality recorded of χαλκός with a quality of copper. To establish Deal as the landing-place of Cæsar all material circumstances must be shown to agree. If the modern Wroxeter is the ancient Uriconium, there must be the like agreement of all features of the country not subject to alteration by time. Such identities must be expressed in the form A = B. We may say Colour of Pacific Ocean = Colour of Atlantic Ocean. Smell of rotten egg = Smell of hydrogen sulphide. In these and similar propositions we assert identity of single qualities or causes of sensation. In the same form we may also express identity of any group of qualities, as in χαλκός = Copper. Deal = Landing-place of Cæsar. A multitude of propositions involving singular terms fall into the same form, as in The Pole star = The slowest-moving star. Jupiter = The greatest of the planets. The ringed planet = The planet having seven satellites. The Queen of England = The Empress of India. The number two = The even prime number. Honesty = The best policy. In mathematical and scientific theories we often meet with simple identities capable of expression in the same form. Thus in mechanical science “The process for finding the resultant of forces = the process for finding the resultant of simultaneous velocities.” Theorems in geometry often give results in this form, as Equilateral triangles = Equiangular triangles. Circle = Finite plane curve of constant curvature. Circle = Curve of least perimeter.