their difference; then the sum of their squares varies as their product. If any quantity vary as another, any power or root of the former will vary, as a like power or root of the latter. If any quantity vary as the product of two others, and if one of the latter be considered constant, the first will vary as the other. If one quantity varies as another, the former is equal to the product of the latter into some constant quantity.

ARITHMETICAL AND GEOMETRICAL PROGRESSION.--Quantities which increase or decrease by a common difference are in arithmetical progression. When they increase, they form an ascending series, and when they decrease they form a descending series. In an arithmetical progression, the last term is equal to the first, the product of the common dif. ference into the number of terms less one. In an ascending series, the first term is the least, and the last the greatest. In a descending series, the first term is the greatest, and the last term the least. In arithmetical progression, the sum of the extremes, is equal to the sum of any other two terms equally distant from the extremes. The sum of the terms is equal to half the sum of the extremes multiplied into the number of terms. In the series of odd numbers, the sum of the terms is always equal to the number ef terms. If there be two ranks of quantities in arithmetical progression, the sum or difference will also be in arithmetical progression. If all the terms of an arithmetical progression be multiplied or divided by the same quantity, the products or quotients will be in arithmetical progression. Quantities are in geometrical progression, when they increase by a common multiplier, or decrease by a common divisor. In geometrical progression, the last term is equal to the product of the first, into that power of the ratio whose index is one less than the number of terms. The sum of a series in geometrical progression, is found by multiplying the last term into the ratio, subtracting the first term, and dividing the remainder by the ratio less one. Quantities in geometrical progression are proportional to their differences.

INFINITES AND Infinitesimals.-A mathematical quantity is said to be infinite, when it is supposed to be increased beyond any determinate limits. When a quantity is dimin shed till it becomes less than any determinate quantity, it is called an infinitesimal. For all practical purposes an infi

nitesimal may be considered as absolutely nothing. If a finite quantity be multiplied by an infinitesimal, the product will be an infinitesimal. If a finite quantity be divided by an infinitesimal, the quotient will be infinite. If a finite quantity be divided by an infinite quantity, the quotient will be an infinitesimal.

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To find the greatest common measure.-Divide one of the quantities by the other, and the preceding divisor by the last remainder, till nothing remains the last divisor will be the greatest common measure. The binominal theorem.-The index of the leading quantity of the power of a binominal, begins in the first term with the index of the power, and decreases regularly by one. The index of the following quantity begins with one in the second term and increases regu. larly by one. The co-efficient of the first term is one; that of the second is equal to the index of the power; and uni versally, if the co-efficient of any term be multiplied by the index of the leading quantity in that term, and divided by the index of the following quantity increased by one, it will give the co-efficient of the succeeding term. A residual quantity may be involved in the same manner, without any variation, except in the signs. To extract the roots of compound quantities, take the root of the first term, for the first term of the required root. Subtract the power from the given quantity, and divide the first term of the remainder by the first term of the root, involved to the next inferior power, and multiplied by the index of the given power; the quotient will be the next term of the root. Subtract the power of the terms already found from the given quantity, and using the same divisor, proceed as before.

GEOMETRY.

BOOK I.-DEFINITIONS.

1. A point is that which has position, but not magnitude. 2. A line is length without breadth. Cor. The extremities of a line are points; and the intersections of one line with another are also points. 3. If two lines are such that they

cannot coincide in any two points without coinciding altogether, each of them is called a straight line. Cor. Hence two straight lines cannot enclose a space. Neither can two

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straight lines have a common segment; that is, they cannot coincide in part without coinciding altogether. 4. A superficies is that which has only length and breadth. Cor. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines. 5. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 6. A plane rectilinial angle is the inclination of two straight lines to one another, which meet together but are not in the same straight line. 7. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. 8. An obtuse angle is that which is greater than a right angle. 9. An acute angle is that which is less than a right angle. 10. A figure is that which is enclosed by one or more boundaries.-The word area denotes the quantity of space contained in a figure, without any reference to the nature of the line or lines which bound it. A circle is a plain figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. 12. And this point is called the centre of the circle. 13. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 14. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. 15. Rectilineal figures are those which are contained by straight lines. 16. Trilateral figures, or triangles, by three straight lines. 17. Quadri. lateral, by four straight lines. 18. Multilateral figures or polygons, by more than four straight lines. 19. Of three sided figures, an equilateral triangle is that which has three equal sides. 20. An isosceles triangle is that which has only two sides equal. 21. A scalene triangle, is that which has three unequal sides. 22. A right angled triangle, is that which has a right angle. 23. An obtuse angled triangle, is that which has an obtuse angle. 24. An acute angled triangle, is that which has three acute angles. 25. Of

four sided figures, a square is that which has all its sides equal, and all its angles right angles. 26. An oblong, is that which has all its angles right angles, but has not all its sides equal. 27. A rhombus, is that which has its sides equal, but its angles are not right angles. 28. A rhomboid, is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. 29. All other four sided figures besides these, are called Trapezi. ums. 30. Parallel straight lines, are such as are in the same plane, and which being produced ever so far both ways, do not meet.

AXIOMS.

1. Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are doubles of the same thing, are equal to one another. 7. Things which are halves of the same thing, are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. 9. The whole is greater than its part. 10. All right angles are equal to one another. 11. "Two straight lines which intersect one another, cannot be both parallel to the same straight line."

PROP. IV. If two triangles have two sides of the one equal to two sides of the other, each to each; and have like. wise the angles contained by those sides equal to one another, their bases, or third sides, shall be equal; and the areas of the triangles shall be equal; and their other angles shall be equal, each to each, viz, those to which the equal sides are opposite. V. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles upon the other side of the base shall also be equal. COR. Every equilateral triangle is also equiangular. VI. If two angles of a triangle be equal to one another, the sides which subtend, or are opposite to them, are also equal to one another. VII. Upon the same base, and on the same side of it, there cannot be two triangles, that

have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity, equal to one another. VIII.

If two triangles have two sides of the one equal two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides of the other. XIII. The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. XIV. If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines are in one and the same straight line. XV. If two straight lines cut one an. other, the vertical, or opposite angles are equal. COR. 1. From this it is manifest, that if two straight lines cut one another, the angles which they make at the point of their intersection, are together equal to four right angles. 2. And hence, all the angles made by any number of straight lines meeting in one point, are together equal to four right angles. XVI. If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles. XVII. Any two angles of a triangle are together less than two right angles. XVIII. The greater side of every triangle has the greater angle opposite to it. XIX. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it. XX. Any two sides of a triangle are together greater than the third side. XXI. If from the ends of one side of a triangle, there be drawn two straight lines to a point within the triangle, these two lines shall be less than the other two sides of the triangle, but shall contain a greater angle. XXIV. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of the one greater than the angle contained by the two sides of the other; the base of that which has the greater angle shall be greater than the base of the other. XXV. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle contained by the sides of that which has the greater base, shall be greater than the angle contain. ed by the sides of the other. XXVI. If two triangles have