angular space will be the fame. And this is alfo the principle by which we can change spaces from one shape to another as will be obvious to every one who understands the forty fifth propofition. I fhall conclude this differtation, with a general admonition to the reader, to observe particularly the use of the supposition in every propofition; because I know nothing that is so easily neglected the construstion and even the references to other propofitions may make a forcible impreffion upon the mind; but the fuppofition has hardly any thing which can awaken our attention to it, except a want of evidence in the conclufion, and there are those who have a wonderful inclination to lay afide their fcruples at that part of a demonstration. DISSERTATION IV. WH WHOEVER has read the first book with care, will perceive the particular use of the triangle for discovering and connecting the different properties of magnitudes; and must at the fame time be fenfible of the uncommon genius of the author, who could apply this figure in such a manner as to make it discover its own properties, as well as those of other magnitudes. In his fecond book he presents us with a new inftrument, the use of which is no less extensive than that of the triangle; and it is the purpose of this differtation to draw the attention of the superficial reader to the properties and most obvious uses of the rectangle. CHA P. I. Of parallelograms. TO understand this fecond book properly, it will be neceffary to confider the thirty fifth propofition of the first book, in more points of view than are immediately presented to us by the propofition itself. It is there demonstrated that when the parallelograms are upon the fame base, and between the fame parallels, the parallelogram spaces are always equal: now this reduces an infinite variety of parallelograms to one fingle parallelogram as far as the fpace is concerned; and if a choise of a parallelogram is to be made, which is to represent all the others, the rectangle, or rightangled parallelogram ought to have the preference; because its bounding lines are the leaft, and its angle fixt or determinate; that its fides are the least will be obvious, by comparing a fide of · VOL. I. 1 the the rectangle with the fide of any other of this infinite variety of parallelograms juft mentioned; as for inftance AB with BE, the one of which fubtends an acute angle and the other a right angle in the fame triangle. But this rectangle itself, is fufficiently or rather abfolutely determined by any two of the lines about any one of its angles: and thus two straight lines, will fix an infinite variety of parallelogram spaces. Now this is the first definition of the fecond book, which is thus expreffed; Every right angled parallelogram, is faid to be contained by any two of the straight lines containing the right angle. The space indeed is bounded by four straight lines; but not by four different straight lines, for the oppofite fides are the fame, or equal. Here the reader is carefully to observe that there is no difference made here between equality and identity for whatever lines are equal, in the prefent cafe they are always confidered asi the fame; that is if you make a right angled parallelogram, any two straight lines whatever, that are equal to the two lines about one of the angles; are faid to contain the space, as well as the par-: ticular lines which make two fides of the very figure. This circumftance perplexes beginners exceedingly; and ought to be made the subject of frequent meditation before one attempts to read a fingle propofition in this book. And here it would be proper to begin with confidering the nature of a parallelogram, ftill more particularly: the reader has feen what will fix the parallelogram space: he is next to confider what will fix the whole parallelogram, fides and angles; and he will firft obferve that the two fides about any of the angles; will determine all the fides; because the other two fides are equal to these two; next let him try to find out what will determine the angles. Now any angle of a-parallelogram being given, it is impoffible to change any of the others; but this the reader will do well to demonftrate; which he may do as follows; any two angles of a parallelogram which follow each other in order are equal to two right angles, one of these being fixed furely the other is fixed alfo; and the oppofite angles of parallelograms are equal; therefore it is obvious that one angle of the parallelogram fixes all the others. The reader should de monftrate this formally and then he will fee that the parallelogram is entirely fixed when an angle, and the two fides about that angle are given. But as I know from experience that the ftudent will not perceive this immediately; let him take only the two fides, and the angle between them; and with these compleat a variety of parallelograms; and he will always find them the fame. Not that this conftruction is neceffary towards a right understanding of the conclufion; but I know that it is neceffary for fixing the reader's attention; and without attention it is impoffible to comprehend any conclufion. Now this point being once settled, it is very clear, that if the parallelogram be right angled, the two fides about one of the angles, or even any lines, equal to these, will fix the parallelegram; from which the full import of this definition may be understood. Containing fome remarks on the principles made use of in demonftrating the first eight propofitions. THE fimplicity of the demonftrations made ufe of in the first eight propofitions, will be readily acknowledged by every one who understands them: and yet it often happens that the ftudent is at a lofs to comprehend their meaning, efpecially upon the first reading; which is owing to two circumftances; first because he does not understand the first definition; and fecondly because we have no common notions of the properties of figures contained in this book; if fome property of a triangle fhould be accidentally mentioned to a person ignorant of the principles of this fcience, he would nevertheless form fome opinion concerning its truth or falfhood from fome common notions of his own; but read the enunciation of the third propofition in this book to him; and he will have no more opinion of its truth or falfhood than if it were delivered to him in an unknown tongue. And fuppofing these two difficulties to be got over; that is fuppofing the first definition to be well understood, and the meaning of the propofition become familiar, it is impoffible for any evidence to present itself to the mind 12 i mind in a fimpler form, than that which thefe demonftrations offer; ; as the progrefs of it in general is thus; you describe a cer tain rectangle or square, which is divided into right angled parallelograms or fquares by lines drawn parallel to the fides; and the whole mystery confifts in fettling what these spaces are according to the first definition. For instance, in the first propofition, as the whole is equal to its parts, the rectangle, or right angled parallelogram, BH is equal to the rectangles BK, DL, EH taken all together; fo that we have only to settle what the rectangles BH, BK, DL, EH are according to the first definition, and the demonftration is compleat. In the fame manner in the second propofition (which by the bye is only a particular case of the first,) the square ABED is made up of the two rectangles AF and CE; fo that we have only to determine what these are according to the first definition, and then the thing proposed is demonstrated. Again in the third propofition, the rectangle AE is made up of the rectangle AD and the square CE; fo that we have only to determine what thefe are according to the fame first definition, and the propofition is demonstrated and fo in other inftances. The first eight propofitions feem to a fuperficial reasoner to rest in fome measure upon a kind of intuitive evidence; because the form of demonftration differs remarkably from that made use of in the remaining propofitions of this book; in which the conclufions are inferred from the forty seventh propofition of the first book without exhibiting the squares. But Euclid never draws any confequence from what we are supposed to fee, though we ourfelves may for his demonftrations are equally conclufive whether the figures upon which we reafon be the objects of our fenfes or not. The different spaces exhibited in these figures arife from the construction, and have their existence and properties from that, whether we see them or not. CHA P. III. Of the addition and substraction of rectangles and Squares: WHEN the learner has made himself master of the demonftrations contained in this fecond book; it will be proper to turn his thoughts |