# Leibniz Legible — https://leibnizlegible.com # Page: https://leibnizlegible.com/page/806933011:0792 # Permalink to this reading: https://leibnizlegible.com/page/806933011:0792?run=74 # Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71) # Folio - , page id 806933011:0792 # Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=806933011&page=792 # Source image: https://digitale-sammlungen.gwlb.de/content/806933011/jpgs/default/00000792.jpg # Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 128 lines, status machine # Lines: 128 recognised, mean confidence 0.88 # Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German. # Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text. # Wording rule: say "the machine reads it as …", never "Leibniz wrote". # After the first blank line: the recognised lines in reading order, one per line. V. or. Suppose von vould Square chis Surd, : 22; because the Index 4, is compounded ol 2 anc 23 V: 12. is chetrue Froduct, or rhe rrue Square Olthe Surd ROot Y: 12. 50 20 theSquare OF V: 10 i5 V: 1C The Realon ot ivhich, is plain: For suxpose cho : 16 = 2 Were t0 be squared; irs square in Surds Will be expressed thus, V: 16 == 4 Bur woten a simple Surd Ouanciry, whosc Radi- cal sign hach For ies Index some Pernaer Number oreater chan 3, 25 6, 9, (9c. And tis required t0 Involre rhis Surd Cubically. Then only pretix betore the Ouantiey a Radical Sign, wich an In dex, wtiich is one chiird ot cheformer, and ris done Thus, IE V: 64, were 10 be Cubed, ic will bC V: 64, and the Cube oF V: 522. 15 V: 512, 00 al(orhe Biquadrare oF V: 5. I5 25 (25 being thC Square ot che Square ol Y: 5.) And ihe Cube ol V: 81, Wvill V: 8Y OF 9. In the general, tosquare, Cube, Rc. any Surd Root, is ontr ro Square or Cube che Power, re cainins che lame NoreoF Radicalier; bur ris vet ter where ic can be donc, tocake one halF, pare 59C. oF ete FXponent oFthe ROOr, as is abore Mewn inthe last particular Rules. Onthe contrary, it vou vould extract che Square Cubo, Or other Roor ot any Surd, rou inust double oxtriple, (090. che Exponene ot che Radicalier. Thus che square Root OF Y: 16 is Y: 16. che Square R00 oC V: 27161: 27. 90. Division of Simple Surd RootS. I. IF che Surds are Similar, Flomosencal, Or0 the Came kind, divide onc Number, Or Ouantity by anocher, and prelix che common Radical Sisn coche Ouorient: Buris cher are Lererogencal, Ot noc oE che same tcind, ehes mult be reduced besore they can be divided. Thus, V: 9d Y: 326 (V: 64 = 8. And s: 5 C 35 (Y: 2 The Demonstration oF wotich General Rule, i the Camie as thac in Mulciplication; for from che Nacure ot Division, che Diritor is toDnity:: a ihe Dividend eo che Ouotient. Therctore in our sirst Instance, V: 9. V: l: : V.: 576. V: 64 buc 25 chese Roors are, (0 Will cheir Squares bc That iS, 9. I : : 576. 64. and chac chese Number are cruly Proportional, is apparene; because the Rectaneles oF Fhe FXtreamS and MeanS are equal WVheretore, V: 9. V: F: : V: 576. V: 64. anC consequemts 64 is che true Quorienr. TI. YF any Racional Ouanrier be ro be dirided b itS Square ROOT, the Square RoOrwill BetheOuo. tient: For it a b bc dirided by V: a b, the Onori ene mustbe V: ab: And if 50 bc dirided by v 50, the Ouoriene Will al(obc V: 50. AIOit, any Rational Ouancier berobe dirided by a Surd, cha Racional Ouanciry must sirst be reduced 10 cho Form oF 2 Surd, bY Rule F. LII. Dhena Surd RoOE havins a Rarional Ouan- cir prefix d belore Il, 15 lo be divided oy lne Surd parr oFic, the duoriene will bethe Rational Ouan- Eier. ThuS, il 5 V: 9, betobedirided by V: 9 the Ouoriene must be 5: A5 it 5 V: 9 had beei dirided br5. che Duorienc voula be V: 3. IV. Wrhen elie Diridend and Divisorare che Pro- ces et C Macteut Ludnririe; mulriplied seve Fally into one common Surd; Or wlien cher are Cctclus 6) Cl Lultotat Pdet et CC DLIOE, atic ertiar reluls, is the ue Suortel. LMuS, IF 8 V: s bc dirided b7 2 V: 5; chc Ouorienr will be 4, and if 8 Y: 7, be divided by 4V: 2, che Ouoriene will be Onif 2. V. Buc vohen che Diridend and Dirilor are rvo Racional anciries, or Numbers, prefix d io rwo unequal Surds; rhen ron must diride, noc only 25 betore, che Rational parr oF the Diridend By chat Ot che Dirilor, buc also che Surd part; and chose eivo Cuorienrs connected rogetter, (0 as the Ratio- nal pari Rand on the Lefe Lland, are the erue Juo: tient Coughr. Thus, il 4 V: 15 vere co be dirided by 2 V: 5. the Ouotiene Will be2 V: 3 (= V: 12.) and il 4V: 12, Were 10 be dirided by3. J: 2, cheu0- tiete will BC V: 6. T. YVHeN HNO Ox more Simple ana Lguat sutas are to be added, mulciply one oF chem by che Number oF them all, and the Product is the SuMI required. Thus, The Sum oF V: 5, and V: 5, is che V: 203 because V: 5 mulciplied by 2, che Number ofche Surds, chat is by V: 4, sires V: 20 cheie Sum. AlCo che Sum OF Y: 7 F Y: 3 V: 7; becauseche Surds are 3 in Number, i5 s: 189; Because Y: 7 mulciplied b7 3 (1. c.) che Y: oc 27 mates Y : 189. TI. BUE iF 2 nequat Simple Surds OF the lame tind are ro be added cogether, or it orie ro be sub eracted From the other, vou must sirst ery Wwhethee cherare Commenturable; and it cher De, chatis. it wten cher hare been dirided by their grearest common Divisor, cheie Ouorients are Rarionat Ouancicies, then rou must Mulciply the Sum ob chose Rationas Ouanciries by che laid Common Dirisor, and the Produet will be the Sum ol the Surds propoled: Or il chie Ditterence oF chose Ra- rional Ouorients be multiplied by the Common Dirisor, Thenthe Product willbe the Difference ot che siren Surds, when che lels is ratcen trom che Sreater. Thus, IF che Sum or Diserence OF chele tvc Surds, V: 50, and v: 8. Were required; becaule they are unequal, L ery sirst, Whether cheY ArC Commenturable, or nor, by dividing cach by the sreatest common Dirilor V: 2. And che Ouotients are V: 25 and V: 4; chac i5, 5 and 2, Wtich are Racional Numbers; and cheretore che Surds arc Commenturable: Then cheir Sum 7, or their Dit ference 3, mulciplied by che common Diriser S: 2 Produces 7 V: 2 For che Sum, and 3 V : 2 EOF che Difierence ot che durds required. TIT. TEche Commensurable Surds proxoled, hac been Fractione, oe Mrine Wuribere, reduced re che Focu Racional Ouancules Pren d Detore one Comnion Surd; then diride ehe Rarional parr oF the Diri- Aadition and Subtraction os Surd Roots.