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. 2. [1] +Die Quersumme+: sum across = sum of the digits. [2] +also+: therefore; _never_ translate it by _also_. [3] +geradstellig, ungeradstellig+: even, odd. [4] +ein Mehrfaches+: a multiple. [5] +der Nenner, der Zähler+: denominator, numerator. [6] see 1 Note 9. [7] +Grundfaktoren+: prime factors. [8] +d. h. (das heisst)+: that is to say (_i. e._) [9] +an und für sich+: by itself, _per se_. [10] +beziehungsweise+: or as the case may be. [11] +darin, dass man ... dividiert+: in dividing (_lit.:_ in this, namely that we divide). +Darin+ anticipates and represents the following clause. [12] +der beiden betreffenden Zahlen+: of the two numbers in question. [13] +das Glied, Vorderglied, Hinterglied+: term, antecedent, consequent. 3. [1] +soll erhoben werden+: is to be raised. Very frequently +soll+ with an infinitive means _is to_. [2] +das Quadrat+: square; +das Biquadrat+: fourth power. [3] +vereinfachen+: to simplify. [4] +es sei x+: let x be. [5] +beträgt (betragen)+: amounts to. [6] +die Rechnung stimmt+: the calculation is correct. [7] +belaufen sich+: amount to. [8] +die Beschaffenheit+: nature. [9] +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. [10] See 1 Note 2; +zieht ... ab (abziehen)+: subtracts. 4. [1] +Soll+; see 3 Note 1. [2] +und zwar+: and moreover. [3] +gerade so viel+: just as much. [4] +überdies+: in addition, to boot. [5] +sieben Meilen zurückgelegt+: makes 7 miles. The German mile varied in different sections from 4.7 to 5.6 U. S. miles. [6] +zu Stande bringen+: do, accomplish, finish. [7] +in eben der Zeit+: in the same time. [8] +genötigt+: obliged. [9] +herabzusetzen+: to reduce. [10] +Angaben+: data, statements. [11] +die Mandel+: lot or set of 15. [12] +um 306+: by 306. +Um+ often means _to the amount of, to the extent of_. [13] +das Merkmal+: characteristic. [14] +nach dem Berichte+: according to the report. [15] +das erste Feld+: the first square. [16] +wenn ... auch+: even if. 5. [1] +z. B. (zum Beispiel)+: for example. [2] +die ... Linien+; see 1 Note 10. [3] +einerlei Seite+: one and the same side. [4] +kongruent+: equal in every respect. [5] +betragen+; see 3 Note 5. [6] +Es+ is merely introductory; the subject is +Seiten+. [7] +man stecke ... ab+: lay off; _lit._ let one lay off. [8] +also+; see 2 Note 2. [9] +alsdann = dann+: then. [10] +daran stossenden (Seite)+: side adjacent to it. 6. [1] +eine ... Figur+; see 1 Note 10. [2] +die ... senkrechte+; see 1 Note 10. [3] +die Sehne+: chord.