"The centre of gravity of any hemisphere [is on the straight line which] is its axis, and divides the said straight line in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3."

"Any segment of a right-angled conoid (i.e., a paraboloid of revolution) cut off by a plane at right angles to the axis is 1½ times the cone which has the same base and the same axis as the segment"

"The centre of gravity of any cone is [the point which divides its axis so that] the portion [adjacent to the vertex is] triple [of the portion adjacent to the base]."

"The centre of gravity of any cylinder is the point of bisection of the axis."

"δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω. [Dôs moi pâ stô, kaì tàn gân kinásō.]"

"First then I will set out the very first theorem which became known to me by means of mechanics, namely that Any segment of a section of a right angled cone (i.e., a parabola) is four-thirds of the triangle which has the same base and equal height, and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]..."

"I am persuaded that it [The Method of Mechanical Theorems] will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me."

"I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards... But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge."

"It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively."

"In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side."

"The centre of gravity of a parallelogram is the point of intersection of its diagonals."

"The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides."

"Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes."

"If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity."

"Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance."

"Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible."

"How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!"

"Noli turbare circulos meos. or Noli tangere circulos meos."