Observe that this sentence begins with a verb, but is not a question, nor a command; also that the next clause begins with so. Under these circumstances supply it at the beginning.
[2]Wievielmal ... sovielmal: as many times ... so many times.
[3]Before translating um, see if the phrase ends with zu and an infinitive; if so um = in order.
[4]seinem Werte nach: according to its value = as regards its value.
[6]Jede Art der Einheiten: each kind of units.
[7]indem man ... rückt and dadurch, dass man ... rückt are two German ways of saying by moving. Verbs following indem man or dadurch, dass man should be turned into the English present participle with by.
[9]die ... Stelle: the place. This is the compound adjective construction which is so common in scientific German. Observe how the phrase is built up:
| die werdende Stelle | the becoming place |
| die leer werdende Stelle | the empty-becoming place |
| die dadurch leer werdende Stelle | the thereby empty becoming place |
The article (die) or some other determining word is separated from its noun (Stelle) by a number of words; but in all cases the word next before the noun is an adjective or participle (werdende), which in turn is preceded by a word qualifying it (leer) and so on. In English the corresponding words follow the noun in the reverse order. This note will be frequently referred to.
[10]Die Quersumme: sum across = sum of the digits.
[1]also: therefore; never translate it by also.
[2]geradstellig, ungeradstellig: even, odd.
[3]der Nenner, der Zähler: denominator, numerator.
[5]see 1 Note 9.
[6]d. h. (das heisst): that is to say (i. e.)
[8]an und für sich: by itself, per se.
[9]beziehungsweise: or as the case may be.
[10]darin, dass man ... dividiert: in dividing (lit.: in this, namely that we divide). Darin anticipates and represents the following clause.
[11]der beiden betreffenden Zahlen: of the two numbers in question.
[12]das Glied, Vorderglied, Hinterglied: term, antecedent, consequent.
[13]soll erhoben werden: is to be raised. Very frequently soll with an infinitive means is to.
[1]das Quadrat: square; das Biquadrat: fourth power.
[2]beträgt (betragen): amounts to.
[5]die Rechnung stimmt: the calculation is correct.
[6]der Rest verhält sich zur Summe wie eins zu zwei: the remainder is to the sum as one is to two. Das Verhältnis: ratio, relation.
[9]See 1 Note 2; zieht ... ab (abziehen): subtracts.
[10]Soll; see 3 Note 1.
[1]überdies: in addition, to boot.
[4]sieben Meilen zurückgelegt: makes 7 miles. The German mile varied in different sections from 4.7 to 5.6 U. S. miles.
[5]zu Stande bringen: do, accomplish, finish.
[6]in eben der Zeit: in the same time.
[7]um 306: by 306. Um often means to the amount of, to the extent of.
[12]nach dem Berichte: according to the report.
[14]das erste Feld: the first square.
[15]z. B. (zum Beispiel): for example.
[1]die ... Linien; see 1 Note 10.
[2]einerlei Seite: one and the same side.
[3]kongruent: equal in every respect.
[4]betragen; see 3 Note 5.
[5]Es is merely introductory; the subject is Seiten.
[6]man stecke ... ab: lay off; lit. let one lay off.
[7]also; see 2 Note 2.
[8]daran stossenden (Seite): side adjacent to it.
[10]eine ... Figur; see 1 Note 10.
[1]die ... senkrechte; see 1 Note 10.
[2]dazu gehörigen: belonging to it.
[4]ganz beliebig ... gegebene Punkte: points given entirely at pleasure = any given points whatever.
[5]der Centriwinkel: angle at the centre.
[6]der Peripheriewinkel: angle at the cirumference.
[7]inhaltsgleich equal in area; der Inhalt: contents.
[8]die Kathete: leg of a right-angled triangle.
[9]Sei CAB ein ... Dreieck: let CAB be a triangle.
[10]der ... Quadrate; see 1 Note 10.
[11]es lässt sich zeigen: it allowes itself to be shown = it may be shown. A common use of lassen.
[12]Hülfslinien: auxiliary lines, construction lines.
[13]see 1 Note 9.
[14]d. i. (das ist): that is to say (i. e.)
[15]nach keiner Seite hin: in no direction.
[1]wie weit ... auch: however far.
[2]gleichlaufend: German equivalent for parallel. In many cases German uses a foreign word and also a native word for the same term.
[3]die ... Seiten; see 1 Note 10.
[4]verhalten sich: are to each other.
[6]Die ... Senkrechte; see 1 Note 10.
[7]es kommt nur darauf an: it is only required.
[9]Um um: the first um means in order and belongs to the infinitive zu beschreiben; the second um means around.
[1]eines Vielecks; see 1 Note 10.
[4]bis ins Unendliche: to infinity.
[7]der Körper: body; (in Geometry) solid.
[8]zwei ... Ebenen; see 1 Note 10.
[9]die Grundfläche: base (surface).
[11]inhaltsgleich: of equal (cubic) contents = of equal volume.
[1]die Mantelfläche: curved surface.
[2]Ein ... Abschnitt; see 1 Note 10.
[4]Wie verhalten sich: in what ratio are ... to each other.
[5]einem ... Denkmale; see 1 Note 10.
[6]das ... Normalmeter; see 1 Note 10.
[4]Allgemeine Eigenschaften: universal properties.
[5]selbstverständlich: of course.
[8]das Beharrungsvermögen: capacity for persisting = inertia.
[1]sich gerade befindet: just finds itself = just happens to be.
[2]das Kausalgesetz: law of causation.
[3]ein ... Körper; see 1 Note 10.
[4]die Festigkeit: firmness = strength.
[5]reicht ... hin (hinreichen): suffices.
[6]der Annahme widersetzt: opposes the assumption.
[7]infolgedessen (in Folge dessen): in consequence of that.
[8]Beispiele is the object of bieten, and für governs Drehungsbewegungen.
[10]entsprechend: corresponding, appropriate.
[11]äussert sich: manifests itself.
[1]see 1 Note 10.
[2]übt ... aus (ausüben): exerts.
[4]also; see 2 Note 2.
[5]giebt ... an (angeben): states.
[8]Pyknometer: specific gravity flask.
[9]Es sei nun P1 das Gewicht: now let P1 be the weight.
[10]einer ... Marke; see 1 Note 10.
[12]Orte und Lagen: places and positions.
[1]see 1 Note 2.
[2]fortschreitende Bewegung: motion of translation.
[4]die ... Wege: the paths; see 1 Note 10.
[5]von einem ... begriffenen Körper: of a body engaged; see 1 Note 10.
[6]see 1 Note 2.
[7]der ... Weg: the path; see 1 Note 10
[8]infolge (in Folge): in consequence.
[9]einer beliebig: of any desired, of any whatever.
[10]kommt ... zu stande: comes about, is brought about.
[11]bei: in the case of (not by.) Bei is of very common occurence with this meaning, which will generally suggest the proper preposition (at, with, in etc.) to use in English.
[12]die Gewichtseinheit: unit of weight.
[1]am Meeresspiegel: at the level of the sea.
[4]am bequemsten: most conveniently.
[5]see 1 Note 2.
[8]Observe that the comma is used as a decimal point and the period as a sign of multiplication.
[9]also; see 2 Note 2.
[10]nimmt ... zu (zunehmen): increases.
[11]see 1 Note 2.
[12]die Gesammtarbeit: total work.
[14]See 1 Note 2.
[1]bei Verminderung: on diminishing; see 13 Note 12.
[3]musste: had to. Do not translate by must; it is in the past tense.
[5]geleistet: (leisten): performed. Die Leistung: performance.
[6]streng genommen: taken strictly = strictly speaking.
[7]eine Feder spannen: compress a spring.
[8]die Spannkraft: tension, potential.
[9]bei Beurtheilung: in judging; see 13 Note 12.
[10]ist massgebend: is decisive, is a criterion.
[11]u. s. f. (und so fort): and so forth.
[13]noch so komplizierten: never so complicated, no matter how complicated.
[2]rollen: the hyphen indicates that hinab belongs also to rollen: to roll down.
[3]stellt ... dar (darstellen): represents.
[4]einen ... Druck; see 1 Note 10.
[5]aufheben: to neutralize, balance.
[6]entstanden (entstehen): originated, formed.
[9]bez. (beziehungsweise): or as the case may be.
[11]der Schraubengang: turn of the screw.
[13]die Bolzenstärke: thickness of the cylinder. Stärke, usually strength, also means thickness.
[15]die Kernstärke: thickness, of the core.
[16]See 1 Note 2., arbeitet ... aus: works out, hollows out.
[17]der Betrag, um den: the amount to the extent of which....
[18]die Steigung oder Ganghöhe: pitch.
[19]See 2 Note 2.
[20]Stellschrauben: set screws; stellen, to adjust.
[21]verhalten sich umgekehrt: they are to each other inversely.
[1]hierbei: in this case (not hereby), Compare 13 Note 12.
[3]findet ... statt (stattfinden): takes place.
[4]bei Klingelzügen: in bell-pulls.
[5]wobei: in which; compare 13 Note 12.
[6]der Flaschenzug: block and tackle.
[8]das Rad an der Welle: wheel and axle.
[9]einem ... Seile; see 1 Note 10.
[10]Riemen- und Seilscheiben: pulleys for belts and ropes.
[11]die Fortpflanzung: propagation, transmission.
[1]einer ... Flüssigkeit: of a liquid; see 1 Note 10.
[2]infolgedessen (in Folge dessen): in consequence of which.
[3]pflanzt sich ... fort (sich fortpflanzen): propagates itself.
[4]Ein ... Körper: a body; see 1 Note 10.
[5]sogen. (sogenannte): so-called.
[6]Ist ... taucht; see 1 Note 2
[7]dabei: in that case; compare 13 Note 12.
[8]eines ... Gefässes: of a vessel; see 1 Note 10.
[10]zweischenkelig: two-legged, two-armed; der Schenkel, thigh.
[4]dabei: at the same time (not thereby).
[5]die Festigkeit: strength (of materials).
[6]das Mass: measure (do not confound with die Masse, mass).
[7]die Zugfestigkeit: tensile strength.
[8]der Querschnitt: cross-section.
[9]die Grösse: magnitude, quantity.
[10]die rückwirkende Festigkeit: compressive strength.
[11]die relative Festigkeit: transverse strength.
[12]Grössen- und Formänderungen: changes of size and form. The hyphen indicates that änderungen belongs to Grössen also.
[13]geschmeidig: plastic, pliable.
[17]die Gehörempfindung: sensation of hearing.
[1]die Wellenbewegung: wave motion.
[2]die Fortpflanzungsgeschwindigkeit: velocity of propagation.
[3]das Fadentelephon: string telephone (toy).
[4]Zwei ... Stücke; see 1 Note 10.
[5]vorwiegend: preponderating; translate: mostly.
[8]Sind; see 1 Note 2.
[9]Sinusschwingungen: sine oscillations.
[10]hängt ... ab (abhängen): depends.
[11]bei; see 13 Note 12.
[13]der Grundton: fundamental note.
[14]das Gepräge: coinage = character.
[15]die Klangfarbe: tone color = timbre.
[16]