CHA P. III. Geometrical Arithmetick. An Introductive Problem. Between two Right Lines, A 4, and B 9, to find a Mean Propor Defini tion. A tional Line. Mean Proportional Line between two other Lines, is fuch a Line whofe Length being multiplied in it felf, fball produce fuch a Number as fhall be equal to the Products of the lengths of the two given Lines, they being multiplied one by the other. Raw a Right line at pleasure, as the line CF, then take the length of the given line A 4 in your Compaffes, and fer it from F to E; alfo take the other given line B9 in your Compaffes, and fet it from E to C; fo is the line CF equal to both the given lines A 4 and B 9, (the point of joining them together Tbeing in the point E.) This done, divide the line CF into two equal parts in D, and upon D, as a Center, with the distance DC or D F, defcribe the Semicircle CGF: Laftly, from the point of joining of the two given lines A and B, namely, from the point E, erect the Perpendicular E G, cutting the Semicircle in G; fo fhall the line E G be a Mean Proportional between the two given Lines A and B, and will contain 6 fuch parts, as the whole line C F contains 13, that is, as A 4, and B 9: And fo this Mean Proportional Line E G, 6, multiplied in it felf will produce 36, equal to the line A 4, multiplied into the line B 9, for 9 times 4 is 36 also. TAHO R E H i REDUCTION. I. How to reduce a long Square or Parallelogram A B C D, into lelogram, as A C, from B to D; alfo take the shortest fide A B, and fet that from D to H; and upon the point of joining at D, erect Perpendicular: Then divide the line BH into two equal parts in K, and upon K (as a Center) defcribe the Semicircle BFH, cutting the Perpendicular D F in F; fo fhall DF be the fide of the Geometrical Square, GEFD, which shall be equal in quantity to the long Square A B CD. For, Suppose the leffer fide of the Long Square A B to be 8 (of any meafure) and the longer fide A C to be 18, the Mean Proportional between them F D will be 12; and fo 18 multiplied by 8, the Product will be 144; and fo will 12 multiplied by 12, produce the fame. L II. To Reduce a Triangle into a Geometrical Square. ET LMN be a Triangle given to be reduced into a Square. Draw a line at pleasure, as LR, then take the length of the Bafe of the Triangle LN, and fet it from L to N;then take half the length of the Perpendicular PO, and fet it from N to R, and upon N erect the Perpendicular NQ;then divide the line LR into two equal parts at S, and L T S upon S as a Center, with the distance S L or SR, describe the Semicircle L QR, cutting the Perpendicular N Qin Q; fo fhall QN be the fide of a Geometrical Square, equal to the Triangle MLN. NR III. Te III. To Reduce a Circle into a Geometrical Square. B IV. Two Triangles of different heights being given, to reduce them to Ꭺ D one height. L' ET there be two Triangles A B C, and CE F, and let it be required to reduce the Triangle E C F into another Triangle that shall be equal in height with the Triangle ABC. First, Through the point A draw a line A D, parallel to the bases of the two Triangles, viz. BF; then extend the fide E F of the Triangle CEF, till it cut the Parallel line in D,and draw DC; then through the point E draw the line EG parallel to DC. Laftly, draw the line DG, fo fhall you have a new Triangle D G F, equal to CE F, and of equal height with the Triangle ふ V. How to Reduce an irregular Figure of Four fides into a Triangle, from an Angle given. LET LT TT HKLN be a Tropezia, or a Figure of Four unequal Sides and Angles, and let it be required to reduce the fame into a Triangle, by a Line drawn from the Angle N. H X N. Extend the fide KL, which is oppofite to the Angle N, towards M, and from N draw a line to the oppofite Angle M, as N K; then through the point H draw the line H M parallel to NK: Laftly, draw the line NM, fo fhall you have a Triangle LMN, equal to the irregular Figure H K L M for the Triangle V, left out in the irregular Figure, is equal the Triangle X, which is taken into the Triangle. M K VI. To reduce an irregular Figure of Four fides into a Triangle, by a line drawn from a Point limited in any of the fides thereof. L' ET ABCD be an irregu lar figure, and it is required to reduce the fame into a Triangle, by lines drawn from the point E, in one of the fides thereof. From the given Point E draw two lines to the two oppofite Angles, as the lines EC and E D, and extend the fide CD, oppofite to the given point E, out on both fides towards F and G; then through the point A draw the line AF, parallel to EC, and the line BG paralle to ED, cutting the fide CD (extended) in the Points F and G: Laftly, draw the two lines E F and E G; fo fhall you have a Triangle EFG, equal to the irregular figure A B C D. VII. To Reduce an irregular figure of Five fides, into a Triangle, by lines drawn from any of the Angles thereof. ET OPQRS be the figure given, and let it be required to Reduce it into a Triangle, by lines drawn from the Angle at O. First, Prolong the fide QR, oppofite to the given Angle at O, both ways, towards T and I; and from the Angle O draw the lines OQ and OR; alfo through the Angular point P and S, draw the lines P T and SI, parallel to O'Q and S R, cutting the fide QR (extended) in the points T and I: Laftly, Draw the lines OT 4 R 1 and and OI; fo fhall you have a Triangle O TI, equal to the five fided figure O P Q R S VIII. To Reduce a Geometrical Square into a Figure of a Lunary Form.. F LET ET ABCD be the Square given; first draw the Diagonal line A C, and on the end thereof at Cerect the Perpendicular E C,making it equal to AC; then continue the fide of the Square A B to E; and on B, as a Center, with the distance B A or B E, defcribe the Semicircle AFE. Laftly, On the point C, and diftance C A, describe the Arch A G E, leaving the Lunary Figure A FEG, equal to the Geometrical Square A B C D. D ADDITION. I. To add two Geometrical Squares together, and to give the Sum of them together in one Square. G L ET the two lines A and B be the fides of the two Geometrical Squares, the one being 30, the other 40, of any Measure. First, Join two lines DC and EC together, making a right angle at C: Then take the line B in your Compaffes, and set it from C to D; alfo take the line A in the Compaffes, and fet that from C to E, and draw the line ED, fo fhall ED, be the fide of the Square DE FG, which fhall be equal to both the Squares made of the lines A and B. For the Line A being 40, the Square made thereof will be 160; and the Line B being 30, the Square made thereof will be 90; which added together, make 250; and fo the fide E D being 50, the Square thereof is 250, equal to both the other. |