1.1. Extract from Third Edition:

In order to determine the position of a point in space, some fixed point is taken for the origin of coordinates, and through it are drawn three fixed planes, called the coordinate planes, at right angles to one another, and intersecting one another two and two in straight lines, which are also at right angles to one another, and are called the axes of the coordinates. Then, if the perpendicular distances of a point from each of the coordinate planes be given, its position will be completely determined.

2.1. Extract from Second Edition:

The Integral Calculus is the reverse of the Differential, and has for its object to determine the value of a function the differential coefficient of which is known, in the same manner as the object of the latter is to determine the differential coefficient when the function itself is given; or, more generally, the object of the Integral Calculus is to discover the relations which exist between the variables and their functions, from given equations between the variables, the functions, and the differential coefficients of the functions. ...

Every rule given in the Differential Calculus for finding the differential coefficient of a function of one variable, being inverted, will furnish a corresponding rule for integration.

This the Integral Calculus, at least in the simpler parts of the subject, requires no new investigation of principles, but depends for them entirely upon the Differential Calculus; and to a person who is familiar with the latter, it offers few difficulties beyond those arising from complicated algebraical operations. Expertness in performing these, and in foreseeing to what result any substitution will lead, is very necessary in this subject; since, with all the rules that can be given, the integration of many formulas may be facilitated, and sometimes can only be effected by particular transformations and artifices, which the student must himself discover.

3.1. Extract from Second Edition:

In order to determine the position of a point in a plane, some fixed point in the plane is taken for the origin of coordinates; and through it are drawn two fixed lines, called the coordinate axes. at right angles to one another.

Then, if the perpendicular distance (to which the name ordinate is given) of a point from each of the coordinate axes be assigned, its position will be completely determined.

4.1. Extract from Second Edition:

The effect the general solution of equations, would be to find the expression for all the roots of an equation of any assigned degree in terms of its coefficients, the coefficients being general symbols. This has hitherto been done only for equations of a degree not exceeding the 4th; and even for cubic equations, it will be seen that the functions of the coefficients which express the roots, are insufficient to give the numerical values of the roots when they are all real; hence we are led to suppose that if we could obtain general formulas for the roots of equations of the 5th and superior degrees, we should be unable to obtain from them the numerical values of the roots by the simple substitution of the numerical values of the coefficients. It has therefore become necessary to invent methods for obtaining, either exactly or approximately, the roots of numerical equations; and which, although only applicable to such equations, depend for their demonstration upon certain general properties of equations, which it is the object of the following Articles to exhibit. The investigations constituting the Theory of Equations, which may in general be conducted by processes purely algebraical and elementary, besides affording a knowledge which will have its use in almost every branch of Analysis, will be found particularly serviceable as an exercise to the mathematical student.

4.2. Extract from Third Edition:

[It] has been confirmed by successive investigations of Geometers ... that the general solution of an equation whose coefficients are indeterminant, and whose roots have no particular relation to one another, is impossible beyond the fourth degree. It has also been shown by Abel that, if two roots are so connected that one of them can be expressed rationally in terms of the other, the equation, if its degree be a prime number, is solvable by radicals; and if a composite number, its solution depends on that of equations of inferior degrees. And more recently Galois has demonstrated that, in order that an irreducible equation, whose degree is a prime number, may be solvable by radicals, it is necessary and sufficient that, any two of its roots being given, the others can be obtained rationally from them.

5.1. Extract from the Second Edition:

A differential equation is said to be of $n$th order, when the differential coefficient of the highest order which it involves is the $n$th.

A differential equation of any order is said, moreover, to be the first, second, &c, degree, when the differential coefficient which marks its order, is raised to the first, second, &c, power: or when it involves a product of at most m dimensions in differential coefficients and their powers, it is said to be of the m th degree

To integrate a differential equation of any order, is to pass to the primitive equation between the variables and the constants, from which the proposed may have been derived by the process of differentiation.

6.1. Extract from the Second Edition:

1. Astronomy is that branch of Natural Philosophy, which treats of the apparent and real motions of the heavenly bodies. In that division of the science which forms the subject of this Treatise, both of these are investigated from observation and calculation, and in Physical Astronomy, the latter are accounted for on mechanical principles.
   
   
2. The starry heavens appear, to a spectator on the Earth's surface, to be a vast spherical vault, of which his eye forms the centre, and in which the heavenly bodies are situated. It is certain that this appearance is fallacious, and that the stars are not all at the same distance from us; only the eye, being unable to judge of the distance of very remote bodies, and having no scale to compare it with, supposes them all equally distant. When therefore for the purpose of explanation we speak of the celestial vault in which the stars are fixed, we mean an imaginary spherical surface of vast dimensions concentric with the eye of the spectator, to which the places of the stars are referred by lines drawn from them to the eye of the spectator.
   
   
3. The Earth, at first sight, appears to one of its inhabitants to be a plane, extended indefinitely on all sides, and sustaining the celestial vault; but it is soon perceived that this cannot be the case, and that, as the Sun and the other heavenly bodies pass under it, the Earth must have a limit both laterally and also downwards; and its round figure, besides being suggested by the analogous forms of the Sun and Moon, is immediately inferred from the following obvious considerations. 1st, When a ship approaches land from any quarter, the topmast first becomes visible to a spectator on the shore, and the hull comes in sight last, which, being the largest part, ought to be first seen, if the Earth were a plane. 2nd, The surface of the sea, when viewed from the deck of a ship or the mast-head, is not seen to lose itself in distance and mist, but to be terminated by a clear, sharp, well-defined circle having the spectator in the centre, exactly such as would be swept out on the surface of a sphere by a line touching it and passing through a fixed point a little elevated above the surface of the sphere. 3rd, In lunar eclipses, when the Earth is interposed between the Sun and Moon, the Earth's shadow upon the Moon's disk appears round, in all positions of the Earth; and lastly, the Earth has been sailed round in various directions. We conclude therefore that, like the other heavenly bodies, the Earth has the figure of a globe (differing very little from a sphere, as will be seen) every where isolated in space, and surrounded by the heavens.
   
   
4. By far the greater part of the heavenly bodies, though collectively in motion, seem never to change their relative situation, that is, their distance from one another; whence they have been called Fixed Stars. The angle which the line joining any two of them subtends, is the same in all parts of the Earth; from which it is evident that the distance of any two points on the Earth's surface is evanescent, compared with the distance of the fixed stars from the Earth, and that the Earth is in reality but a point in space. This causes a spectator, wherever he is placed, to find himself in the centre of the celestial movements; and therefore, instead of supposing the centre of the superficies in which fixed stars appear to lie, to be the eye of the spectator, we may suppose it to be the centre of the Earth.

7.1. Extract:

The boundary of every plane section of a sphere is a circle.

It the cutting plane pass through the centre, this is evident; and in this case the section is called a great circle, and is determined when any two points on the surface of the sphere through which it passes are given. All great circles are equal to one another, since they have the same radius, namely that if the sphere; and they all bisect one another, since their planes intersect in diameters of the sphere. hence the distance of the points of intersection of two great circles measured on the sphere is a semi-circumference.

If the cutting plane does not pass through the centre of the sphere, from $O$ the centre, drop upon it the perpendicular $OC$, and join $C$ with any point $A$ in the boundary of the section; then

          $\sqrt{AO^2 - OC^2}$, which is invariable;

therefore the boundary of the section is a circle whose centre is $C$; and its is called a small circle. Arcs of small circles are rarely used; and when hereafter an arc of a circle is mentioned, an arc of a great circle, unless the contrary be specified, is invariably intended, and in most cases it is employed to denote the angle which it subtends at the centre of the sphere, no regard being had to the radius of the sphere.

8.1. Extract from Second Edition:

In any plane triangle there are six parts to be considered, three angles, and three sides. In order to find all the rest, it is in general sufficient to know three of them, but one of these must be a side; because with three given angles (provided their sum be equal to two right angles) we can form an infinite number of triangles, which are not equal, but only similar to one another. Geometry furnishes simple constructions for each of these cases in which we can determine a triangle by means of some of its parts; but these constructions, on account of the imperfection of the instruments employed, give only a rough and often insufficient approximation. Mathematicians have therefore sought to substitute for them numerical calculations, which always attain the required degree of exactness.

The special object of Trigonometry is to give methods for calculating all the parts of a triangle when there are sufficient data; this is what is called solving a triangle. but in its present enlarged sense, Trigonometry treats of the principles by which angular magnitudes may be estimated, and numerically connected with one another, and with other magnitudes; and shows how to perform measurements generally, by means of the relations of the sides and angles of rectilinear figures.