this+value+is+much+used

² = helpful, useful, profitable, handy [handier -comp., handiest -sup.].

Ex. If concepts in the A/Z subject index entries were listed in the same citation order employed in the classification of these subjects the result would be much less helpful.

Ex. The subject approach is also useful for nonfiction, but has little value for fiction.

Ex. With this type of facility search strategy can be refined to give the most profitable output.

Ex. The volumes can be used as handy desk references.

----

* acabar + Posesivo + vida útil = run towards + the end of + Posesivo + useful life.

* agotar + Posesivo + vida útil = run towards + the end of + Posesivo + useful life.

* área útil = floor area.

* dejar de ser útil = outlive + Posesivo + usefulness.

* hacerse útil = come into + Posesivo + own.

* metros útiles = square footage.

* muy útil = highly valuable.

* poco útil = unhelpful.

* resultar útil = prove + fruitful, hold + Nombre + in good stead, stand + Nombre + in good stead.

* ser útil = hold + Nombre + in good stead, stand + Nombre + in good stead, come in + useful, come into + Posesivo + own.

* ser útil para = be of service to.

* vida útil = shelf life, service life.

* vida útil de un documento = shelf life.

impendo ( inp-), di, sum, 3, v. a. [inpendo], to weigh out, lay out, expend (class.; cf.: insumo, erogo).

Lit.:

accipe inquam, nam hoc inpendit puplicum,

Plaut. Stich. 5, 4, 35:

non erunt tam amentes, ut operam, curam, pecuniam impendant in eas res, quas, etc.,

Cic. Verr. 2, 4, 30, § 68 fin.:

nummos in navem,

Dig. 14, 1, 1, § 9:

pecuniam in opsonio, etc.,

ib. 24, 1, 31 fin.:

HS. octogies pro introitu novi sacerdotii,

Suet. Claud. 9:

istuc, quod tu de tua pecunia dicis impensum,

Cic. Verr. 2, 5, 19, § 47:

nescio quid impendit et in commune contulit,

id. Quint. 3, 12:

certus sumptus impenditur,

id. Verr. 2, 3, 98, § 227:

intellegebant, sese sibi et populo Romano, non Verri et Apronio serere, impendere, laborare,

id. ib. 2, 3, 52, §

121: sed quid ego vos, de vestro inpendatis, hortor?

Liv. 6, 15, 9:

quis aegram et claudentem oculos gallinam impendat amico tam sterili,

lay out the value of, Juv. 12, 96.—

Transf., in gen., to expend, devote, employ, apply:

impensurus omne aevi sui spatium in id opus,

Vell. 2, 89:

vitam usui alicujus,

Tac. A. 12, 65:

vitam patriae,

Luc. 2, 382:

vitam famae,

Stat. S. 5, 1, 63:

biennium libris componendis,

Quint. Ep. ad Tryph. 1; cf. Suet. Aug. 98:

tota volumina in hanc disputationem,

Quint. 3, 6, 21:

vim suam in plura,

id. 1, 12, 2:

operam, curam in aliquid,

Cic. Verr. 2, 4, 30, § 68:

nihil sanguinis in socios,

Ov. M. 13, 266:

quid censetis in hoc foedere faciendo voluisse Mamertinos impendere laboris, operae, pecuniae, ne? etc.,

Cic. Verr. 2, 5, 20, § 51:

omnis impendunt curas distendere, etc.,

Verg. G. 3, 124:

hunc oculum pro vobis impendi,

Petr. 1:

quae (studia) juvenibus erudiendis impenderam, Quint. prooem. § 1: omnia studiis,

id. 12, 11, 19; cf.:

tantum laboris studiis,

id. 2, 4, 3; 1, 1, 3:

aliquem exemplo,

to use as a warning, Front. Strat. 4, 1, 33:

verba animi proferre et vitam impendere vero,

Juv. 4, 91.—Hence,

impensus ( inp-), a, um, P. a. (lit., profusely expended; hence), ample, considerable, great.

Lit.:

impenso pretio,

i. e. high, dear, Cic. Att. 14, 13, 5; * Caes. B. G. 4, 2, 2; Liv. 2, 9, 6; for which also absol.:

impenso,

Hor. S. 2, 3, 245.—

Transf., large, great, strong, vehement: dear, expensive:

in his rebus unus est solus inventus, qui ab hac tam impensa voluntate bonorum palam dissideret,

Cic. Sest. 62, 130:

voluntas erga aliquem,

Liv. 35, 44, 3:

libido,

Lucr. 5, 964:

studium,

Fronto, Ep. ad M. Caes. 2, 4 Mai.:

opera,

Gell. 9, 14, 6.— Comp.:

impensior cura,

Ov. M. 2, 405; Tac. H. 1, 31:

verbis laudare,

Val. Max. 4, 3, ext. 1:

injurias atrociores impensiore damno vindicare,

Gell. 20, 1, 32:

vae misero illi, cujus cibo iste factuist impensior,

larger, stouter, fatter, Plaut. Capt. 4, 2, 26:

nam pol ingrato homine nihil impensiu'st,

more expensive, id. Bacch. 3, 2, 10.— Sup.:

preces,

Suet. Tib. 13.—Hence, adv.: im-pensē ( inp-).

At great cost, expensively:

impensius unge, puer, caules,

Pers. 6, 68:

bibliothecas incendio absumptas impensissime reparari curavit,

Suet. Dom. 20. —

Transf., exceedingly, greatly, very much; earnestly, eagerly, zealously (freq.; esp. in the comp.; cf.: magnopere, admodum, perquam, etc.).

With verbs:

illi invidere misere, verum unus tamen impense,

Ter. Eun. 3, 1, 23; cf.:

aliquid impense cupere,

id. Ad. 5, 9, 36:

retinere,

Liv. 40, 35, 7:

petere,

Quint. 10, 5, 18; Suet. Claud. 11:

demirari,

Gell. 9, 9, 15:

atque acriter atque inflammanter facit (odium in Verrem),

id. 10, 3, 13 (this the better read. al. incense).— Comp.:

eo facio id impensius, quod, etc.,

Cic. Fam. 13, 64, 1:

agere gratias,

Liv. 37, 56, 10:

consulere,

Verg. A. 12, 20:

venerari numina,

Ov. M. 6, 314:

instare,

id. ib. 7, 323:

crescere his dignitas, si, etc.,

Liv. 1, 40, 2:

accendi certamina in castris,

id. 4, 46, 2.—

With adjj.:

impense improbus,

Plaut. Ep. 4, 1, 39:

impense doctus,

Gell. 13, 10, 4.—

impensa ( inp-), ae, f. (sc. pecunia), outlay, cost, charge, expense (cf.: sumtus, impendium).

Lit. (class.; in sing. and plur.):

impensam ac sumptum facere in culturam,

Varr. R. R. 1, 2, 8:

quoniam impensam fecimus in macrocola,

Cic. Att. 13, 25, 3:

nullam impensam fecerant,

id. Phil. 6, 5, 19:

arationes magna impensa tueri,

id. Verr. 2, 3, 21, § 53:

columnae nulla impensa dejectae,

id. ib. 2, 1, 55, §

145: sine impensa,

id. Rep. 2, 14:

exigua,

Suet. Vesp. 18:

publica,

id. Claud. 6:

matris ac vitrici,

id. Tib. 7:

sua,

Plin. 36, 5, 4, § 42:

nostra,

Ov. H. 7, 188:

quia inpensa pecuniae facienda erat,

Liv. 44, 23, 1:

haec nimia est inpensa,

Juv. 12, 97:

finem impensae non servat prodiga Roma,

id. 7, 138:

parcere impensae,

to economize, id. 5, 156.—In plur.:

atque etiam impensae meliores, muri, navalia, portus, aquarum ductus, etc.,

Cic. Off. 2, 17, 60:

ludorum ac munerum,

Suet. Tib. 34:

operum ac munerum,

id. Dom. 12:

itineris,

id. Vit. 7:

cenarum,

Hor. Ep. 1, 19, 38:

publicae,

Tac. H. 4, 4; Plin. 12, 14, 32, § 63:

nolo meis impensis illorum ali luxuriam,

i. e. of my reputation, Nep. Phoc. 1, 4:

inpensas conferre,

to contribute to expenses, Juv. 3, 216. —

Transf. (so perh. not ante-Aug.).

In gen.:

cruoris,

Ov. M. 8, 63:

operum,

Verg. A. 11, 228:

officiorum,

Liv. 37, 53, 12. —

In partic., that which is used up or expended for any purpose, materials, ingredients; for repairing an aqueduct (timber, stone, earth, etc.), Front. Aquaed. 124;

of the stuffing for sausages, etc.,

Arn. 7, 231;

of sacrifices,

Petr. 137;

of masonry,

Pall. 1, 13.

impendo ( inp-), di, sum, 3, v. a. [inpendo], to weigh out, lay out, expend (class.; cf.: insumo, erogo).

Lit.:

accipe inquam, nam hoc inpendit puplicum,

Plaut. Stich. 5, 4, 35:

non erunt tam amentes, ut operam, curam, pecuniam impendant in eas res, quas, etc.,

Cic. Verr. 2, 4, 30, § 68 fin.:

nummos in navem,

Dig. 14, 1, 1, § 9:

pecuniam in opsonio, etc.,

ib. 24, 1, 31 fin.:

HS. octogies pro introitu novi sacerdotii,

Suet. Claud. 9:

istuc, quod tu de tua pecunia dicis impensum,

Cic. Verr. 2, 5, 19, § 47:

nescio quid impendit et in commune contulit,

id. Quint. 3, 12:

certus sumptus impenditur,

id. Verr. 2, 3, 98, § 227:

intellegebant, sese sibi et populo Romano, non Verri et Apronio serere, impendere, laborare,

id. ib. 2, 3, 52, §

121: sed quid ego vos, de vestro inpendatis, hortor?

Liv. 6, 15, 9:

quis aegram et claudentem oculos gallinam impendat amico tam sterili,

lay out the value of, Juv. 12, 96.—

Transf., in gen., to expend, devote, employ, apply:

impensurus omne aevi sui spatium in id opus,

Vell. 2, 89:

vitam usui alicujus,

Tac. A. 12, 65:

vitam patriae,

Luc. 2, 382:

vitam famae,

Stat. S. 5, 1, 63:

biennium libris componendis,

Quint. Ep. ad Tryph. 1; cf. Suet. Aug. 98:

tota volumina in hanc disputationem,

Quint. 3, 6, 21:

vim suam in plura,

id. 1, 12, 2:

operam, curam in aliquid,

Cic. Verr. 2, 4, 30, § 68:

nihil sanguinis in socios,

Ov. M. 13, 266:

quid censetis in hoc foedere faciendo voluisse Mamertinos impendere laboris, operae, pecuniae, ne? etc.,

Cic. Verr. 2, 5, 20, § 51:

omnis impendunt curas distendere, etc.,

Verg. G. 3, 124:

hunc oculum pro vobis impendi,

Petr. 1:

quae (studia) juvenibus erudiendis impenderam, Quint. prooem. § 1: omnia studiis,

id. 12, 11, 19; cf.:

tantum laboris studiis,

id. 2, 4, 3; 1, 1, 3:

aliquem exemplo,

to use as a warning, Front. Strat. 4, 1, 33:

verba animi proferre et vitam impendere vero,

Juv. 4, 91.—Hence,

impensus ( inp-), a, um, P. a. (lit., profusely expended; hence), ample, considerable, great.

Lit.:

impenso pretio,

i. e. high, dear, Cic. Att. 14, 13, 5; * Caes. B. G. 4, 2, 2; Liv. 2, 9, 6; for which also absol.:

impenso,

Hor. S. 2, 3, 245.—

Transf., large, great, strong, vehement: dear, expensive:

in his rebus unus est solus inventus, qui ab hac tam impensa voluntate bonorum palam dissideret,

Cic. Sest. 62, 130:

voluntas erga aliquem,

Liv. 35, 44, 3:

libido,

Lucr. 5, 964:

studium,

Fronto, Ep. ad M. Caes. 2, 4 Mai.:

opera,

Gell. 9, 14, 6.— Comp.:

impensior cura,

Ov. M. 2, 405; Tac. H. 1, 31:

verbis laudare,

Val. Max. 4, 3, ext. 1:

injurias atrociores impensiore damno vindicare,

Gell. 20, 1, 32:

vae misero illi, cujus cibo iste factuist impensior,

larger, stouter, fatter, Plaut. Capt. 4, 2, 26:

nam pol ingrato homine nihil impensiu'st,

more expensive, id. Bacch. 3, 2, 10.— Sup.:

preces,

Suet. Tib. 13.—Hence, adv.: im-pensē ( inp-).

At great cost, expensively:

impensius unge, puer, caules,

Pers. 6, 68:

bibliothecas incendio absumptas impensissime reparari curavit,

Suet. Dom. 20. —

Transf., exceedingly, greatly, very much; earnestly, eagerly, zealously (freq.; esp. in the comp.; cf.: magnopere, admodum, perquam, etc.).

With verbs:

illi invidere misere, verum unus tamen impense,

Ter. Eun. 3, 1, 23; cf.:

aliquid impense cupere,

id. Ad. 5, 9, 36:

retinere,

Liv. 40, 35, 7:

petere,

Quint. 10, 5, 18; Suet. Claud. 11:

demirari,

Gell. 9, 9, 15:

atque acriter atque inflammanter facit (odium in Verrem),

id. 10, 3, 13 (this the better read. al. incense).— Comp.:

eo facio id impensius, quod, etc.,

Cic. Fam. 13, 64, 1:

agere gratias,

Liv. 37, 56, 10:

consulere,

Verg. A. 12, 20:

venerari numina,

Ov. M. 6, 314:

instare,

id. ib. 7, 323:

crescere his dignitas, si, etc.,

Liv. 1, 40, 2:

accendi certamina in castris,

id. 4, 46, 2.—

With adjj.:

impense improbus,

Plaut. Ep. 4, 1, 39:

impense doctus,

Gell. 13, 10, 4.—

impensa ( inp-), ae, f. (sc. pecunia), outlay, cost, charge, expense (cf.: sumtus, impendium).

Lit. (class.; in sing. and plur.):

impensam ac sumptum facere in culturam,

Varr. R. R. 1, 2, 8:

quoniam impensam fecimus in macrocola,

Cic. Att. 13, 25, 3:

nullam impensam fecerant,

id. Phil. 6, 5, 19:

arationes magna impensa tueri,

id. Verr. 2, 3, 21, § 53:

columnae nulla impensa dejectae,

id. ib. 2, 1, 55, §

145: sine impensa,

id. Rep. 2, 14:

exigua,

Suet. Vesp. 18:

publica,

id. Claud. 6:

matris ac vitrici,

id. Tib. 7:

sua,

Plin. 36, 5, 4, § 42:

nostra,

Ov. H. 7, 188:

quia inpensa pecuniae facienda erat,

Liv. 44, 23, 1:

haec nimia est inpensa,

Juv. 12, 97:

finem impensae non servat prodiga Roma,

id. 7, 138:

parcere impensae,

to economize, id. 5, 156.—In plur.:

atque etiam impensae meliores, muri, navalia, portus, aquarum ductus, etc.,

Cic. Off. 2, 17, 60:

ludorum ac munerum,

Suet. Tib. 34:

operum ac munerum,

id. Dom. 12:

itineris,

id. Vit. 7:

cenarum,

Hor. Ep. 1, 19, 38:

publicae,

Tac. H. 4, 4; Plin. 12, 14, 32, § 63:

nolo meis impensis illorum ali luxuriam,

i. e. of my reputation, Nep. Phoc. 1, 4:

inpensas conferre,

to contribute to expenses, Juv. 3, 216. —

Transf. (so perh. not ante-Aug.).

In gen.:

cruoris,

Ov. M. 8, 63:

operum,

Verg. A. 11, 228:

officiorum,

Liv. 37, 53, 12. —

In partic., that which is used up or expended for any purpose, materials, ingredients; for repairing an aqueduct (timber, stone, earth, etc.), Front. Aquaed. 124;

of the stuffing for sausages, etc.,

Arn. 7, 231;

of sacrifices,

Petr. 137;

of masonry,

Pall. 1, 13.

impendo ( inp-), di, sum, 3, v. a. [inpendo], to weigh out, lay out, expend (class.; cf.: insumo, erogo).

Lit.:

accipe inquam, nam hoc inpendit puplicum,

Plaut. Stich. 5, 4, 35:

non erunt tam amentes, ut operam, curam, pecuniam impendant in eas res, quas, etc.,

Cic. Verr. 2, 4, 30, § 68 fin.:

nummos in navem,

Dig. 14, 1, 1, § 9:

pecuniam in opsonio, etc.,

ib. 24, 1, 31 fin.:

HS. octogies pro introitu novi sacerdotii,

Suet. Claud. 9:

istuc, quod tu de tua pecunia dicis impensum,

Cic. Verr. 2, 5, 19, § 47:

nescio quid impendit et in commune contulit,

id. Quint. 3, 12:

certus sumptus impenditur,

id. Verr. 2, 3, 98, § 227:

intellegebant, sese sibi et populo Romano, non Verri et Apronio serere, impendere, laborare,

id. ib. 2, 3, 52, §

121: sed quid ego vos, de vestro inpendatis, hortor?

Liv. 6, 15, 9:

quis aegram et claudentem oculos gallinam impendat amico tam sterili,

lay out the value of, Juv. 12, 96.—

Transf., in gen., to expend, devote, employ, apply:

impensurus omne aevi sui spatium in id opus,

Vell. 2, 89:

vitam usui alicujus,

Tac. A. 12, 65:

vitam patriae,

Luc. 2, 382:

vitam famae,

Stat. S. 5, 1, 63:

biennium libris componendis,

Quint. Ep. ad Tryph. 1; cf. Suet. Aug. 98:

tota volumina in hanc disputationem,

Quint. 3, 6, 21:

vim suam in plura,

id. 1, 12, 2:

operam, curam in aliquid,

Cic. Verr. 2, 4, 30, § 68:

nihil sanguinis in socios,

Ov. M. 13, 266:

quid censetis in hoc foedere faciendo voluisse Mamertinos impendere laboris, operae, pecuniae, ne? etc.,

Cic. Verr. 2, 5, 20, § 51:

omnis impendunt curas distendere, etc.,

Verg. G. 3, 124:

hunc oculum pro vobis impendi,

Petr. 1:

quae (studia) juvenibus erudiendis impenderam, Quint. prooem. § 1: omnia studiis,

id. 12, 11, 19; cf.:

tantum laboris studiis,

id. 2, 4, 3; 1, 1, 3:

aliquem exemplo,

to use as a warning, Front. Strat. 4, 1, 33:

verba animi proferre et vitam impendere vero,

Juv. 4, 91.—Hence,

impensus ( inp-), a, um, P. a. (lit., profusely expended; hence), ample, considerable, great.

Lit.:

impenso pretio,

i. e. high, dear, Cic. Att. 14, 13, 5; * Caes. B. G. 4, 2, 2; Liv. 2, 9, 6; for which also absol.:

impenso,

Hor. S. 2, 3, 245.—

Transf., large, great, strong, vehement: dear, expensive:

in his rebus unus est solus inventus, qui ab hac tam impensa voluntate bonorum palam dissideret,

Cic. Sest. 62, 130:

voluntas erga aliquem,

Liv. 35, 44, 3:

libido,

Lucr. 5, 964:

studium,

Fronto, Ep. ad M. Caes. 2, 4 Mai.:

opera,

Gell. 9, 14, 6.— Comp.:

impensior cura,

Ov. M. 2, 405; Tac. H. 1, 31:

verbis laudare,

Val. Max. 4, 3, ext. 1:

injurias atrociores impensiore damno vindicare,

Gell. 20, 1, 32:

vae misero illi, cujus cibo iste factuist impensior,

larger, stouter, fatter, Plaut. Capt. 4, 2, 26:

nam pol ingrato homine nihil impensiu'st,

more expensive, id. Bacch. 3, 2, 10.— Sup.:

preces,

Suet. Tib. 13.—Hence, adv.: im-pensē ( inp-).

At great cost, expensively:

impensius unge, puer, caules,

Pers. 6, 68:

bibliothecas incendio absumptas impensissime reparari curavit,

Suet. Dom. 20. —

Transf., exceedingly, greatly, very much; earnestly, eagerly, zealously (freq.; esp. in the comp.; cf.: magnopere, admodum, perquam, etc.).

With verbs:

illi invidere misere, verum unus tamen impense,

Ter. Eun. 3, 1, 23; cf.:

aliquid impense cupere,

id. Ad. 5, 9, 36:

retinere,

Liv. 40, 35, 7:

petere,

Quint. 10, 5, 18; Suet. Claud. 11:

demirari,

Gell. 9, 9, 15:

atque acriter atque inflammanter facit (odium in Verrem),

id. 10, 3, 13 (this the better read. al. incense).— Comp.:

eo facio id impensius, quod, etc.,

Cic. Fam. 13, 64, 1:

agere gratias,

Liv. 37, 56, 10:

consulere,

Verg. A. 12, 20:

venerari numina,

Ov. M. 6, 314:

instare,

id. ib. 7, 323:

crescere his dignitas, si, etc.,

Liv. 1, 40, 2:

accendi certamina in castris,

id. 4, 46, 2.—

With adjj.:

impense improbus,

Plaut. Ep. 4, 1, 39:

impense doctus,

Gell. 13, 10, 4.—

impensa ( inp-), ae, f. (sc. pecunia), outlay, cost, charge, expense (cf.: sumtus, impendium).

Lit. (class.; in sing. and plur.):

impensam ac sumptum facere in culturam,

Varr. R. R. 1, 2, 8:

quoniam impensam fecimus in macrocola,

Cic. Att. 13, 25, 3:

nullam impensam fecerant,

id. Phil. 6, 5, 19:

arationes magna impensa tueri,

id. Verr. 2, 3, 21, § 53:

columnae nulla impensa dejectae,

id. ib. 2, 1, 55, §

145: sine impensa,

id. Rep. 2, 14:

exigua,

Suet. Vesp. 18:

publica,

id. Claud. 6:

matris ac vitrici,

id. Tib. 7:

sua,

Plin. 36, 5, 4, § 42:

nostra,

Ov. H. 7, 188:

quia inpensa pecuniae facienda erat,

Liv. 44, 23, 1:

haec nimia est inpensa,

Juv. 12, 97:

finem impensae non servat prodiga Roma,

id. 7, 138:

parcere impensae,

to economize, id. 5, 156.—In plur.:

atque etiam impensae meliores, muri, navalia, portus, aquarum ductus, etc.,

Cic. Off. 2, 17, 60:

ludorum ac munerum,

Suet. Tib. 34:

operum ac munerum,

id. Dom. 12:

itineris,

id. Vit. 7:

cenarum,

Hor. Ep. 1, 19, 38:

publicae,

Tac. H. 4, 4; Plin. 12, 14, 32, § 63:

nolo meis impensis illorum ali luxuriam,

i. e. of my reputation, Nep. Phoc. 1, 4:

inpensas conferre,

to contribute to expenses, Juv. 3, 216. —

Transf. (so perh. not ante-Aug.).

In gen.:

cruoris,

Ov. M. 8, 63:

operum,

Verg. A. 11, 228:

officiorum,

Liv. 37, 53, 12. —

In partic., that which is used up or expended for any purpose, materials, ingredients; for repairing an aqueduct (timber, stone, earth, etc.), Front. Aquaed. 124;

of the stuffing for sausages, etc.,

Arn. 7, 231;

of sacrifices,

Petr. 137;

of masonry,

Pall. 1, 13.

impendo ( inp-), di, sum, 3, v. a. [inpendo], to weigh out, lay out, expend (class.; cf.: insumo, erogo).

Lit.:

accipe inquam, nam hoc inpendit puplicum,

Plaut. Stich. 5, 4, 35:

non erunt tam amentes, ut operam, curam, pecuniam impendant in eas res, quas, etc.,

Cic. Verr. 2, 4, 30, § 68 fin.:

nummos in navem,

Dig. 14, 1, 1, § 9:

pecuniam in opsonio, etc.,

ib. 24, 1, 31 fin.:

HS. octogies pro introitu novi sacerdotii,

Suet. Claud. 9:

istuc, quod tu de tua pecunia dicis impensum,

Cic. Verr. 2, 5, 19, § 47:

nescio quid impendit et in commune contulit,

id. Quint. 3, 12:

certus sumptus impenditur,

id. Verr. 2, 3, 98, § 227:

intellegebant, sese sibi et populo Romano, non Verri et Apronio serere, impendere, laborare,

id. ib. 2, 3, 52, §

121: sed quid ego vos, de vestro inpendatis, hortor?

Liv. 6, 15, 9:

quis aegram et claudentem oculos gallinam impendat amico tam sterili,

lay out the value of, Juv. 12, 96.—

Transf., in gen., to expend, devote, employ, apply:

impensurus omne aevi sui spatium in id opus,

Vell. 2, 89:

vitam usui alicujus,

Tac. A. 12, 65:

vitam patriae,

Luc. 2, 382:

vitam famae,

Stat. S. 5, 1, 63:

biennium libris componendis,

Quint. Ep. ad Tryph. 1; cf. Suet. Aug. 98:

tota volumina in hanc disputationem,

Quint. 3, 6, 21:

vim suam in plura,

id. 1, 12, 2:

operam, curam in aliquid,

Cic. Verr. 2, 4, 30, § 68:

nihil sanguinis in socios,

Ov. M. 13, 266:

quid censetis in hoc foedere faciendo voluisse Mamertinos impendere laboris, operae, pecuniae, ne? etc.,

Cic. Verr. 2, 5, 20, § 51:

omnis impendunt curas distendere, etc.,

Verg. G. 3, 124:

hunc oculum pro vobis impendi,

Petr. 1:

quae (studia) juvenibus erudiendis impenderam, Quint. prooem. § 1: omnia studiis,

id. 12, 11, 19; cf.:

tantum laboris studiis,

id. 2, 4, 3; 1, 1, 3:

aliquem exemplo,

to use as a warning, Front. Strat. 4, 1, 33:

verba animi proferre et vitam impendere vero,

Juv. 4, 91.—Hence,

impensus ( inp-), a, um, P. a. (lit., profusely expended; hence), ample, considerable, great.

Lit.:

impenso pretio,

i. e. high, dear, Cic. Att. 14, 13, 5; * Caes. B. G. 4, 2, 2; Liv. 2, 9, 6; for which also absol.:

impenso,

Hor. S. 2, 3, 245.—

Transf., large, great, strong, vehement: dear, expensive:

in his rebus unus est solus inventus, qui ab hac tam impensa voluntate bonorum palam dissideret,

Cic. Sest. 62, 130:

voluntas erga aliquem,

Liv. 35, 44, 3:

libido,

Lucr. 5, 964:

studium,

Fronto, Ep. ad M. Caes. 2, 4 Mai.:

opera,

Gell. 9, 14, 6.— Comp.:

impensior cura,

Ov. M. 2, 405; Tac. H. 1, 31:

verbis laudare,

Val. Max. 4, 3, ext. 1:

injurias atrociores impensiore damno vindicare,

Gell. 20, 1, 32:

vae misero illi, cujus cibo iste factuist impensior,

larger, stouter, fatter, Plaut. Capt. 4, 2, 26:

nam pol ingrato homine nihil impensiu'st,

more expensive, id. Bacch. 3, 2, 10.— Sup.:

preces,

Suet. Tib. 13.—Hence, adv.: im-pensē ( inp-).

At great cost, expensively:

impensius unge, puer, caules,

Pers. 6, 68:

bibliothecas incendio absumptas impensissime reparari curavit,

Suet. Dom. 20. —

Transf., exceedingly, greatly, very much; earnestly, eagerly, zealously (freq.; esp. in the comp.; cf.: magnopere, admodum, perquam, etc.).

With verbs:

illi invidere misere, verum unus tamen impense,

Ter. Eun. 3, 1, 23; cf.:

aliquid impense cupere,

id. Ad. 5, 9, 36:

retinere,

Liv. 40, 35, 7:

petere,

Quint. 10, 5, 18; Suet. Claud. 11:

demirari,

Gell. 9, 9, 15:

atque acriter atque inflammanter facit (odium in Verrem),

id. 10, 3, 13 (this the better read. al. incense).— Comp.:

eo facio id impensius, quod, etc.,

Cic. Fam. 13, 64, 1:

agere gratias,

Liv. 37, 56, 10:

consulere,

Verg. A. 12, 20:

venerari numina,

Ov. M. 6, 314:

instare,

id. ib. 7, 323:

crescere his dignitas, si, etc.,

Liv. 1, 40, 2:

accendi certamina in castris,

id. 4, 46, 2.—

With adjj.:

impense improbus,

Plaut. Ep. 4, 1, 39:

impense doctus,

Gell. 13, 10, 4.—

impensa ( inp-), ae, f. (sc. pecunia), outlay, cost, charge, expense (cf.: sumtus, impendium).

Lit. (class.; in sing. and plur.):

impensam ac sumptum facere in culturam,

Varr. R. R. 1, 2, 8:

quoniam impensam fecimus in macrocola,

Cic. Att. 13, 25, 3:

nullam impensam fecerant,

id. Phil. 6, 5, 19:

arationes magna impensa tueri,

id. Verr. 2, 3, 21, § 53:

columnae nulla impensa dejectae,

id. ib. 2, 1, 55, §

145: sine impensa,

id. Rep. 2, 14:

exigua,

Suet. Vesp. 18:

publica,

id. Claud. 6:

matris ac vitrici,

id. Tib. 7:

sua,

Plin. 36, 5, 4, § 42:

nostra,

Ov. H. 7, 188:

quia inpensa pecuniae facienda erat,

Liv. 44, 23, 1:

haec nimia est inpensa,

Juv. 12, 97:

finem impensae non servat prodiga Roma,

id. 7, 138:

parcere impensae,

to economize, id. 5, 156.—In plur.:

atque etiam impensae meliores, muri, navalia, portus, aquarum ductus, etc.,

Cic. Off. 2, 17, 60:

ludorum ac munerum,

Suet. Tib. 34:

operum ac munerum,

id. Dom. 12:

itineris,

id. Vit. 7:

cenarum,

Hor. Ep. 1, 19, 38:

publicae,

Tac. H. 4, 4; Plin. 12, 14, 32, § 63:

nolo meis impensis illorum ali luxuriam,

i. e. of my reputation, Nep. Phoc. 1, 4:

inpensas conferre,

to contribute to expenses, Juv. 3, 216. —

Transf. (so perh. not ante-Aug.).

In gen.:

cruoris,

Ov. M. 8, 63:

operum,

Verg. A. 11, 228:

officiorum,

Liv. 37, 53, 12. —

In partic., that which is used up or expended for any purpose, materials, ingredients; for repairing an aqueduct (timber, stone, earth, etc.), Front. Aquaed. 124;

of the stuffing for sausages, etc.,

Arn. 7, 231;

of sacrifices,

Petr. 137;

of masonry,

Pall. 1, 13.

impendo ( inp-), di, sum, 3, v. a. [inpendo], to weigh out, lay out, expend (class.; cf.: insumo, erogo).

Lit.:

accipe inquam, nam hoc inpendit puplicum,

Plaut. Stich. 5, 4, 35:

non erunt tam amentes, ut operam, curam, pecuniam impendant in eas res, quas, etc.,

Cic. Verr. 2, 4, 30, § 68 fin.:

nummos in navem,

Dig. 14, 1, 1, § 9:

pecuniam in opsonio, etc.,

ib. 24, 1, 31 fin.:

HS. octogies pro introitu novi sacerdotii,

Suet. Claud. 9:

istuc, quod tu de tua pecunia dicis impensum,

Cic. Verr. 2, 5, 19, § 47:

nescio quid impendit et in commune contulit,

id. Quint. 3, 12:

certus sumptus impenditur,

id. Verr. 2, 3, 98, § 227:

intellegebant, sese sibi et populo Romano, non Verri et Apronio serere, impendere, laborare,

id. ib. 2, 3, 52, §

121: sed quid ego vos, de vestro inpendatis, hortor?

Liv. 6, 15, 9:

quis aegram et claudentem oculos gallinam impendat amico tam sterili,

lay out the value of, Juv. 12, 96.—

Transf., in gen., to expend, devote, employ, apply:

impensurus omne aevi sui spatium in id opus,

Vell. 2, 89:

vitam usui alicujus,

Tac. A. 12, 65:

vitam patriae,

Luc. 2, 382:

vitam famae,

Stat. S. 5, 1, 63:

biennium libris componendis,

Quint. Ep. ad Tryph. 1; cf. Suet. Aug. 98:

tota volumina in hanc disputationem,

Quint. 3, 6, 21:

vim suam in plura,

id. 1, 12, 2:

operam, curam in aliquid,

Cic. Verr. 2, 4, 30, § 68:

nihil sanguinis in socios,

Ov. M. 13, 266:

quid censetis in hoc foedere faciendo voluisse Mamertinos impendere laboris, operae, pecuniae, ne? etc.,

Cic. Verr. 2, 5, 20, § 51:

omnis impendunt curas distendere, etc.,

Verg. G. 3, 124:

hunc oculum pro vobis impendi,

Petr. 1:

quae (studia) juvenibus erudiendis impenderam, Quint. prooem. § 1: omnia studiis,

id. 12, 11, 19; cf.:

tantum laboris studiis,

id. 2, 4, 3; 1, 1, 3:

aliquem exemplo,

to use as a warning, Front. Strat. 4, 1, 33:

verba animi proferre et vitam impendere vero,

Juv. 4, 91.—Hence,

impensus ( inp-), a, um, P. a. (lit., profusely expended; hence), ample, considerable, great.

Lit.:

impenso pretio,

i. e. high, dear, Cic. Att. 14, 13, 5; * Caes. B. G. 4, 2, 2; Liv. 2, 9, 6; for which also absol.:

impenso,

Hor. S. 2, 3, 245.—

Transf., large, great, strong, vehement: dear, expensive:

in his rebus unus est solus inventus, qui ab hac tam impensa voluntate bonorum palam dissideret,

Cic. Sest. 62, 130:

voluntas erga aliquem,

Liv. 35, 44, 3:

libido,

Lucr. 5, 964:

studium,

Fronto, Ep. ad M. Caes. 2, 4 Mai.:

opera,

Gell. 9, 14, 6.— Comp.:

impensior cura,

Ov. M. 2, 405; Tac. H. 1, 31:

verbis laudare,

Val. Max. 4, 3, ext. 1:

injurias atrociores impensiore damno vindicare,

Gell. 20, 1, 32:

vae misero illi, cujus cibo iste factuist impensior,

larger, stouter, fatter, Plaut. Capt. 4, 2, 26:

nam pol ingrato homine nihil impensiu'st,

more expensive, id. Bacch. 3, 2, 10.— Sup.:

preces,

Suet. Tib. 13.—Hence, adv.: im-pensē ( inp-).

At great cost, expensively:

impensius unge, puer, caules,

Pers. 6, 68:

bibliothecas incendio absumptas impensissime reparari curavit,

Suet. Dom. 20. —

Transf., exceedingly, greatly, very much; earnestly, eagerly, zealously (freq.; esp. in the comp.; cf.: magnopere, admodum, perquam, etc.).

With verbs:

illi invidere misere, verum unus tamen impense,

Ter. Eun. 3, 1, 23; cf.:

aliquid impense cupere,

id. Ad. 5, 9, 36:

retinere,

Liv. 40, 35, 7:

petere,

Quint. 10, 5, 18; Suet. Claud. 11:

demirari,

Gell. 9, 9, 15:

atque acriter atque inflammanter facit (odium in Verrem),

id. 10, 3, 13 (this the better read. al. incense).— Comp.:

eo facio id impensius, quod, etc.,

Cic. Fam. 13, 64, 1:

agere gratias,

Liv. 37, 56, 10:

consulere,

Verg. A. 12, 20:

venerari numina,

Ov. M. 6, 314:

instare,

id. ib. 7, 323:

crescere his dignitas, si, etc.,

Liv. 1, 40, 2:

accendi certamina in castris,

id. 4, 46, 2.—

With adjj.:

impense improbus,

Plaut. Ep. 4, 1, 39:

impense doctus,

Gell. 13, 10, 4.—

impensa ( inp-), ae, f. (sc. pecunia), outlay, cost, charge, expense (cf.: sumtus, impendium).

Lit. (class.; in sing. and plur.):

impensam ac sumptum facere in culturam,

Varr. R. R. 1, 2, 8:

quoniam impensam fecimus in macrocola,

Cic. Att. 13, 25, 3:

nullam impensam fecerant,

id. Phil. 6, 5, 19:

arationes magna impensa tueri,

id. Verr. 2, 3, 21, § 53:

columnae nulla impensa dejectae,

id. ib. 2, 1, 55, §

145: sine impensa,

id. Rep. 2, 14:

exigua,

Suet. Vesp. 18:

publica,

id. Claud. 6:

matris ac vitrici,

id. Tib. 7:

sua,

Plin. 36, 5, 4, § 42:

nostra,

Ov. H. 7, 188:

quia inpensa pecuniae facienda erat,

Liv. 44, 23, 1:

haec nimia est inpensa,

Juv. 12, 97:

finem impensae non servat prodiga Roma,

id. 7, 138:

parcere impensae,

to economize, id. 5, 156.—In plur.:

atque etiam impensae meliores, muri, navalia, portus, aquarum ductus, etc.,

Cic. Off. 2, 17, 60:

ludorum ac munerum,

Suet. Tib. 34:

operum ac munerum,

id. Dom. 12:

itineris,

id. Vit. 7:

cenarum,

Hor. Ep. 1, 19, 38:

publicae,

Tac. H. 4, 4; Plin. 12, 14, 32, § 63:

nolo meis impensis illorum ali luxuriam,

i. e. of my reputation, Nep. Phoc. 1, 4:

inpensas conferre,

to contribute to expenses, Juv. 3, 216. —

Transf. (so perh. not ante-Aug.).

In gen.:

cruoris,

Ov. M. 8, 63:

operum,

Verg. A. 11, 228:

officiorum,

Liv. 37, 53, 12. —

In partic., that which is used up or expended for any purpose, materials, ingredients; for repairing an aqueduct (timber, stone, earth, etc.), Front. Aquaed. 124;

of the stuffing for sausages, etc.,

Arn. 7, 231;

of sacrifices,

Petr. 137;

of masonry,

Pall. 1, 13.

sestertĭus, a, um, num. adj. [contr. from semis-tertius], two and a half; only in the phrases sestertius nummus and milia sestertia; v. I. A. and I. B. 1. infra.— Mostly as subst.

sestertĭus (written also with the characters HS.; v. B. 4. infra), ii, m. (sc. nummus); also in full: sestertius nummus; gen. plur. sestertiūm; rarely sestertiorum or sestertiūm nummūm, a sesterce, a small silver coin, originally equal to two and a half asses, or one fourth of a denarius. When the as was reduced in weight, during the Punic wars, the denarius was made equal to sixteen asses, and the sestertius continued to be one fourth of the denarius. Its value, up to the time of Augustus, was twopence and half a farthing sterling, or four and one tenth cents; afterwards about one eighth less. The sestertius was the ordinary coin of the Romans, by which the largest sums were reckoned. The sestertium (1000 sestertii) was equal (up to the time of Augustus, afterwards about one eighth less) to

1 d. sterling, or $42.94 in United States coin (v. Zumpt, Gram. § 842; Dict. of Ant. s. v. as, sestertius).

In gen.:

sestertius, quod duobus semis additur (dupondius enim et semis antiquus sestertius est) et veteris consuetudinis, ut retro aere dicerent, ita ut semis tertius, quartus semis pronunciarent, ab semis tertius sestertius dicitur,

Varr. L. L. 5, § 173 Müll.: nostri quartam denarii partem, quod efficie [p. 1686] batur ex duobus assibus et tertio semisse, sestertium nominaverunt, Vitr. 3, 1 med.; Cic. Div. in Caecil. 10, 30:

taxatio in libras sestertii singuli et in penuriā bini,

Plin. 18, 13, 34, § 130.—Freq. joined with nummus:

mille nongentos quinquaginta sestertios nummos,

Col. 3, 3, 9.— Gen. plur. sestertiūm: quid verum sit, intellego;

sed alias ita loquor, ut concessum est, ut hoc vel pro deum dico vel pro deorum, alias, ut necesse est, cum triumvirum non virorum, cum sestertiūm nummūm non nummorum, quod in his consuetudo varia non est,

Cic. Or. 46, 56:

sestertiūm sexagena milia nummūm,

Varr. R. R. 3, 6, 1; cited ap. Plin. 10, 20, 23, § 45.—Rarely, sestertiorum:

duo milia sestertiorum,

Col. 3, 3, 13.—

In partic.

As adj. in neutr. plur., with milia (in Varr. and Col.):

ut asinus venierit sestertiis milibus LX. (= sexaginta milibus sestertium),

Varr. R. R. 2, 1, 14:

grex centenarius facile quadragena milia sestertia ut reddat,

id. ib. 3, 6, 6:

hos numquam minus dena milia sestertia ex melle recipere,

id. ib. 3, 6, 11:

Hirrius ex aedificiis duodena milia sestertia capiebat,

id. ib. 3, 17, 3:

sestertiis octo milibus,

Col. 3, 3, 8; 3, 3, 9; 3, 3, 10.—

To express more than two complete thousands sestertia is used as plurale tantum, with distrib. numerals (rare before the Aug. per.):

si qui vilicus ex eo fundo, qui sestertia dena meritasset... domino XX. milia nummūm pro X. miserit (= decem milia sestertiūm),

Cic. Verr. 2, 3, 50, § 119:

candidati apud eum HS. quingena deposuerunt (= quindecim milia sestertiūm),

id. Att. 4, 15, 7:

capit ille ex suis praediis sexcena sestertia, ego centena ex meis,

id. Par. 6, 3, 49:

bis dena super sestertia nummum,

Hor. Ep. 2, 2, 33:

Tiberius Hortalo se respondisse ait, daturum liberis ejus ducena sestertia singulis,

Tac. A. 2, 38:

princeps capiendis pecuniis posuit modum usque ad dena sestertia,

id. ib. 11, 7.—Rarely with card. numerals:

sestertia centum,

Sall. C. 30, 6:

septem donat sestertia,

Hor. Ep. 1, 7, 80:

centum sestertia,

Mart. 6, 20, 1:

sex sestertia,

id. 6, 30, 1; cf.:

ne cui jus esset nisi qui... HS. CCCC. census fuisset,

Plin. 33, 2, 8, § 32.—

Millions of sesterces were expressed in three ways:

By the words centena (or centum) milia sestertiūm, preceded by a numeral adverb (rare): miliens centena milia sestertium, a hundred millions, etc., Plin. 12, 18, 41, § 84.—

With ellips. of the words centena milia, the gen. plur. sestertiūm being preceded by the numeral adv. (rare;

once in Cic.): HS. (i. e. sestertium) quater decies P. Tadio numerata Athenis... planum faciam (i.e. sestertiūm quater decies centena milia, = 1,400,000 sesterces),

Cic. Verr. 2, 1, 39, § 100 (where B. and K. after Orell. read. ex conj., sestertium; v. Kühner, Gram. § 229, 5 b.).—

With sestertium, declined as subst. neutr., and the numeral adverbs from decies upward (also with ellips. of centena or centum milia; sestertium here = centum milia sestertiūm. The origin of this usage, which became general, has been much disputed, and it is usual to explain it, after Non. p. 495 (cf. Quint. 1, 6, 18), as a grammatical blunder, by which the gen. plur. sestertium has been mistaken for a neutr. sing., Zumpt, Gram. § 873; but it more probably grew out of the adj. use of sestertium with mille, supra; v. Fischer, Gram. 2, p. 269; cf. Neue, Formenl. 1, p. 116; Kühner, Gram. § 209).

Nom. and acc.:

quom ei testamento sestertium milies relinquatur,

Cic. Off. 3, 24, 93:

nonne sestertium centies et octogies... Romae in quaestu reliquisti?

id. Pis. 35, 86:

sestertium sexagies, quod advexerat Domitius,

Caes. B. C. 1, 23, 4:

sestertium quadringenties aerario illatum est,

Tac. A. 13, 31:

sestertium deciens numeratum esse,

Cic. Verr. 2, 2, 7, § 20; 2, 3, 70, § 163:

quadringenties sestertium, quod debuisti,

id. Phil. 2, 37, 93; id. Off. 3, 24, 93; Nep. Att. 14, 2:

sestertium ducenties ex eā praedā redactum esse,

Liv. 45, 43, 8; Val. Max. 9, 1, 6:

sestertium milies in culinam congerere,

Sen. Cons. ad Helv. 10, 3:

quater milies sestertium suum vidit,

id. Ben. 2, 27, 1; Plin. 18, 6, 7, § 37; Tac. A. 6, 45; 12, 22; 12, 53; 13, 31; id. H. 4, 47; Suet. Calig. 37; id. Galb. 5.—Sometimes with ellips. of sestertium:

dissipatio, per quam Antonius septies miliens avertit,

Cic. Phil. 5, 4, 11.—

Gen.:

syngrapha sestertii centies per legatos facta,

Cic. Phil. 2, 37, 95:

argenti ad summam sestertii decies in aerarium rettulit,

Liv. 45, 4, 1:

sestertii milies servus,

Sen. Q. N. 1, 16, 1:

liberalitas decies sestertii,

Tac. A. 2, 37; 2, 86:

centies sestertii largitio,

id. ib. 12, 58; 12, 53; Plin. Ep. 10, 3 (5), 2.—

Abl.:

quadragies sestertio villam venisse,

Varr. R. R. 3, 17, 3:

sexagies sestertio, tricies sestertio,

Val. Max. 9, 1, 4:

centies sestertio cenavit uno die,

Sen. Cons. ad Helv. 9, 11:

pantomimae decies sestertio nubunt,

id. ib. 12, 5; id. Ben. 4, 36, 1; Plin. 8, 48, 74, § 196:

accepto quinquagies sestertio,

Tac. A. 3, 17; 6, 17; 16, 13; id. H. 4, 42; Plin. Ep. 3, 19, 7; Suet. Caes. 50; id. Tib. 48; id. Calig. 38, 4.—The sign HS., i.e. II. and semis, stands for sestertius, sestertia, and sestertium, in all the uses described above; when it is necessary, to avoid ambiguity, its meanings are distinguished thus: HS. XX. stands for sestertii viginti; HS. X̅X̅., with a line over the numeral, = sestertia vicena, or 20,000 sesterces; H̅S̅. X̅X̅., with lines over both signs, = sestertium vicies, or 2,000,000 sesterces (Kühner, Gram. § 229 Anm. 1). But in recent edd. the numerals are usu. written in full, when the meaning would otherwise be doubtful.—

Transf., in gen.

Nummo sestertio or sestertio nummo, for a small sum, for a trifle (good prose):

ecquis est, qui bona C. Rabirii Postumi nummo sestertio sibi addici velit? Tua, Postume, nummo sestertio a me addicuntur,

Cic. Rab. Post. 17, 45; Val. Max. 5, 2, 10:

C. Matienus damnatus sestertio nummo veniit,

Liv. Epit. 55:

quae maxima inter vos habentur, divitiae, gratia, potentia, sestertio nummo aestiman da sunt,

Sen. Ep. 95, 59; Val. Max. 8, 2, 3.—

Money, a sum of money:

sestertio amplo comparare,

for a large sum, Sol. 27 (40) fin. —

In the times of the emperors, also, a copper coin, worth four asses, Plin. 34, 2, 2, § 4; cf. Eckhel. Doctr. Num. 6, p. 283.—

ses-tertĭum, ii, n., in econom. lang., as a measure of dimension, two and a half feet deep:

ipsum agrum sat erit bipalio vertere: quod vocant rustici sestertium,

Col. Arb. 1, 5 (for which:

siccus ager bipalio subigi debet, quae est altitudo pastinationis, cum in duos pedes et semissem convertitur humus,

id. ib. 3, 5, 3).

SUBJECT AREA: Electronics and information technology

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Born into a family of Scottish landowners, at the early age of 13 years Napier went to the University of St Andrews in Fife, but he apparently left before taking his degree. An extreme Protestant, he was active in the struggles with the Roman Catholic Church and in 1594 he dedicated to James VI of Scotland his Plaine Discovery of the Whole Revelation of St John, an attempt to promote the Protestant case in the guise of a learned study. About this time, as well as being involved in the development of military equipment, he devoted much of his time to finding methods of simplifying the tedious calculations involved in astronomy. Eventually he realized that by representing numbers in terms of the power to which a "base" number needed to be raised to produce them, it was possible to perform multiplication and division and to find roots, by the simpler processes of addition, substraction and integer division, respectively.

A description of the principle of his "logarithms" (from the Gk. logos, reckoning, and arithmos, number), how he arrived at the idea and how they could be used was published in 1614 under the title Mirifici Logarithmorum Canonis Descriptio. Two years after his death his Mirifici Logarithmorum Canonis Constructio appeared, in which he explained how to calculate the logarithms of numbers and gave tables of them to eight significant figures, a novel feature being the use of the decimal point to distinguish the integral and fractional parts of the logarithm. As originally conceived, Napier's tables of logarithms were calculated using the natural number e(=2.71828…) as the base, not directly, but in effect according to the formula: Naperian logx= 107(log e 107-log e x) so that the original Naperian logarithm of a number decreased as the number increased. However, prior to his death he had readily acceded to a suggestion by Henry Briggs that it would greatly facilitate their use if logarithms were simply defined as the value to which the decimal base 10 needed to be raised to realize the number in question. He was almost certainly also aware of the work of Joost Burgi.

No doubt as an extension of his ideas of logarithms, Napier also devised a means of manually performing multiplication and division by means of a system of rods known as Napier's Bones, a forerunner of the modern slide-rule, which evolved as a result of successive developments by Edmund Gunther, William Oughtred and others. Other contributions to mathematics by Napier include important simplifying discoveries in spherical trigonometry. However, his discovery of logarithms was undoubtedly his greatest achievement.

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