equivalent to the lune whose angle is A, and which is measured by 2 AXT (P. 15, C, 2). Hence, we have ADE+ AGH=2AXT'; and, for a like reason, BGF+BID=2B XT, and CIH+CFE =2CXT. But the sum of these six triangles exceeds the hemispher, by twice the triangle ABC, and the hemisphere is represented by 4T: therefore, twice the triangle ABC, is equivalent to 2AXT'+2B X T'+2CXT-4T; and, consequently, ABC=-(A+B+C-2)XT; hence, every spherical triangle is measured by the sum of its three angles minus two right angles, multiplied by the tri-rectangular triangle. Scholium 1. When we speak of the spherical angles, we regard the right angle as unity, and compare the sum of the three angles with this standard. Hence, however many right angles there may be in the sum of the three angles minus two right angles, just so many tri-rectangular triangles, will the proposed triangle contain. If the angles, for example, are each equal to f of a right angle, the sum of the three angles is equal to 4 right angles; and this sum, minus two right angles, is represented by 4-2, or 2 ; therefore, the surface of the triangle is equal to two tri-rectangular triangles, or to the fourth part of the surface of the entire sphere. Scholium 2. The same proportion which exists between the spherical triangle ABC, and the tri-rectangular triangle, exists also between the spherical pyramid which has ABC for its base, and the tri-rectangular pyramid. The triedral angle of the pyramid is to the triedral angle of the trirectangular pyramid, as the triangle ABC to the tri-rectangular triangle. From these relations, the following consequences are deduced. First. Two triangular spherical pyramids are to each other as their bases: and since a polygonal pyramid may always be divided into a certain number of triangular pyramids, it follows that any two spherical pyramids are to each other, as the polygons which form their bases. Second. The polyedral angles at the vertices of these pyramids, are also as their bases; hence, for comparing any two polyedral angles, we have merely to place their vertices at the centres of two equal spheres; the angles are to each other as the spherical polygons intercepted between their faces. The vertical angle of the tri-rectangular pyramid is formed by three planes at right angles to each other : this angle, which may be called a right polyedral angle, will serve as a very natural unit of measure for all other polyedral angles. If, for example, the area of the triangle is of the tri-rectangular triangle, the corresponding triedral angle is also of the right polyedral angle. PROPOSITION XIX. THEOREM. The surface of a spherical polygon is equal to the excess of the sum of all its angles, over two right angles taken as many times as there are sides in the polygon less two, multiplied by the tri-rectangular triangle. Let ABCDE be a spherical polygon. From one of the vertices A, let diagonals AC, AD, be drawn to the other vertices; the polygon ABCDE will be divided into as many tri. angles less two, as it has sides. Now, the surface of each triangle is equal to the sum of all its angles less two right angles, into the tri-rectangular triangle. The sum of the angles of all the triangles is the same as that of all the angles of the polygon; hence, the surface of the polygon is equal to the sum of all its angles, diminished by twice as many right angles as it has sides less two, into the tri-rectangu. lar triangle. Scholium. Let s be the sum of all the angles of a spheri cal polygon, n the number of its sides, and T the tri-rectangular triangle; the right angle being taken as unity, the surface of the polygon will be equal to (s-2 (n—2,)) XT=(s—2n+4) RT. APPENDIX. NOTE A.-PAGE 22. A DEMONSTRATION is a train of logical arguments brought to a conclusion. The bases or premises of a demonstration, are definitions, axioms, propositions previously established, and hypotheses. The arguments are the links which connect the premises, logically, with the conclusion or ultimate truth to be proved. In Geometry we employ two kinds of demonstration—. the Direct, and the Indirect or the method involving the Reductio ad absurdum. These are also called Positive and Negative Demonstrations. In the direct method, the premises are definitions, axioms, and previous propositions ; 'and by a process of logical argumentation, the magnitudes of which something is to be proved, are shown to bear the mark by which that may always be inferred, or, in other words, are shown to fall under some definition, axiom, or proposition, previously laid down. The direct demonstration may be divided into two classes : 1st. Where the argument depends on superpositionthat is, on the coincidence of magnitudes when applied the one to the other: and 2dly. Where it depends on addition and subtraction, or immediately on principles previously laid down. The indirect method rests on a hypothesis. This hypothesis is combined in a process of logical argumentation, with definitions, axioms, and previous propositions, until a conclusion is obtained, which agrees or disagrees with some known truth. Now, if the conclusion so deduced, is excluded from the truths previously established, that is, if it is opposed to any of them, then it follows that the hy. pothesis, leading to a result contradictory to such truth, must be false. In the indirect demonstration, therefore, the conclusion is compared with the truths known antecedently to the proposition in question; if it disagrees with any of them, the hypothesis is false. We have examples of the first class of the direct demonstration in the reasoning which establishes Propositions V. and VI.—and of the second class in that which establishes Propositions I. and IV. We have also examples of the indirect method in the demonstrations of Propositions II. and III. It is often supposed, though erroneously, that the indirect demonstration is less conclusive and satisfactory than the direct. This impression is simply the result of a want of proper analysis. For example: in the demonstration of Proposition II. we propose to prove “ that two straight lines having two points in common coincide throughout their whole extent.” Now, it is evident that they either coincide or separate. If they separate, they must separate at some point, as C. But the supposition or hypothesis of their separating at this point, involves the conclusion, that a part is equal to the whole, which is contrary to Axiom 8, and therefore untrue: Hence, they do not separate, and therefore, they coincide. Similar remarks apply to all indi. rect demonstrations. In both kinds of demonstrations the premises and conclusion agree: that is, they are both true or both false, the reasoning or argument in both being supposed strictly logical. For a more full discussion of this subject, see Davies' Logic of Mathematics. THE REGULAR POLYEDRONS. A REGULAR POLYEDRON is one whose faces are all equal regular polygons, and whose polyedral angles are all equal to each other. 1. The TETRAEDRON, or regular pyramid, is a solid bounded by four equal equilateral triangles. 2. The HEXAEDRON, or Cube, is a solid bounded by six equal squares. 3. The OCTAEDRON, is a solid bounded by eight equal equilateral triangles. 4. The DODECAEDRON, is a solid bounded by twelve equal and regular pentagons. 5. The ICOSAEDRON is a solid bounded by twenty equal equilateral triangles. First. If the faces are equilateral triangles, polyedrons may be constructed bounded by such triangles and will have polyedral angles contained either by three, four or five of them: hence arise three regular polyedral bodies, viz: the tetraedron, the octaedron, and the icosaedron, and no others can be constructed with equilateral triangles. For, each angle of an equilateral triangle being equal to a third part of two right, six such angles about the vertex of a polyedral angle would be equal to four right angles, which is impossible (B. VI., P. 20, Secondly. If the faces are squares, their angles may be arranged by threes: hence, results the hexaedron, or cube. Four angles of a square are equal to four right angles, and cannot form a polyedral angle. Thirdly. In fine, if the faces are regular pentagons, their angles likewise may be arranged by threes: the regular dodecaedron will result. We can proceed no farther: three angles of a regular hexagon are equal to four right angles; three of a heptafon are greater. |