Paul Elbourne

The Existence Entailments of Definite Descriptions

Keywords: Definite descriptions, Propositional attitudes, Conditionals, Presupposition

published or not. The documents may come

Introduction

As is well known, [START_REF] Russell | On denoting[END_REF]Whitehead and Russell 1927: 30-32, 66-71) analysed sentences of the form (1) as having the meaning in (2a), which can be paraphrased as in (2b) or (2c).

(1) The φ is ψ.

(2) a. ∃x(φx ∧ ∀y(φy → y = x) ∧ ψx) b. There is an entity such that it is φ and nothing else is φ and it is ψ. c. There is exactly one φ and it is ψ. Heim (1991: 493-94) and [START_REF] Kripke | Russell's notion of scope[END_REF]Kripke ( : 1023) ) and I (Elbourne 2005: 109-12) (henceforth HKE ) have argued that the readings available for certain sentences in which definite descriptions are embedded under propositional attitude verbs and conditionals raise a problem for the claim that the speaker asserts that there is exactly one φ, as opposed to presupposing this. But [START_REF] Kaplan | Reading 'On Denoting' on its centenary[END_REF] 1 985) and Neale (2005: 846;2007: 89-91) (henceforth KN ) dismiss this concern. 1 The purpose of the present article is to show that the argumentation of KN does not vindicate the Russellian analysis on this point. That is, I will argue that there is a real problem here for the Russellian analysis of the definite article. I will begin by recapitulating the objection of HKE and the response of KN; I will then argue that, properly formulated, HKE's objection is not affected by KN's criticism; and I will end by arguing, following [START_REF] Heim | Artikel und Definitheit[END_REF][START_REF] Elbourne | Situations and Individuals[END_REF], that an analysis of the definite article as presuppositional, along the lines of [START_REF] Frege | Über Sinn und Bedeutung[END_REF] and Strawson (1950), captures the relevant facts adequately.

HKE's Objection to Russell

Here, then, is the objection of HKE. We are dealing with examples like the following.

(3) Hans wants the ghost in his attic to be quiet tonight.

(4) Hans wonders whether the ghost in his attic will be quiet tonight.

(5) If the ghost in his attic is quiet tonight, Hans will hold a party.

The Russellian analysis of definite descriptions predicts that (6a), in an appropriate context, should mean (6b).

(6) a. The ghost in his attic will be quiet tonight. b. There is exactly one ghost in Hans's attic and it will be quiet tonight.

But then this seems to predict that (3)-( 5) should have the readings in ( 7)-( 9), respectively.

We simply embed the truth conditions in (6b) under the relevant operators, closely following the syntactic form of the sentences.

(7) Hans wants there to be exactly one ghost in his attic and for it to be quiet tonight.

(8) Hans wonders whether the following is the case: there is exactly one ghost in his attic and it will be quiet tonight.

(9) If there is exactly one ghost in his attic and it is quiet tonight, Hans will hold a party.

The fact is, however, that none of (3)-( 5) have these predicted readings. In saying (3), for example, we would not be attributing to Hans the odd desire to have a ghost in his attic. And in saying (4), we are not saying that Hans wonders, among other things, whether there is exactly one ghost in his attic; it sounds rather as if Hans is assuming that there is exactly one ghost in his attic and wondering only whether it will be quiet tonight. A similar remark can be made about (5): in this example it sounds as if the speaker is assuming that there is exactly one ghost in Hans's attic, whereas this possibility is explicitly left open in (9).

KN's Response to HKE

Thus the objection of HKE. In response, KN point out that propositional attitude verbs do not support entailments. Suppose that we have a sentence of the form S PAs Comp p , where S refers to an agent, PA is a propositional attitude verb, Comp is a complementizer (that or whether ), and p is a declarative sentence. Suppose further that p , uttered in isolation in a particular context c, entails (what we would express by) q , a declarative sentence with distinct truth conditions (as evaluated in c). KN's point is that not all sentences of the form S PAs Comp p entail the corresponding sentence S PAs Comp q . For example, (10) entails ( 11), but [START_REF] May | Logical Form: its Structure and Derivation[END_REF] does not entail (13) (Kaplan 2005: 985).

(10) There are honest men.

(11) There are men. [START_REF] May | Logical Form: its Structure and Derivation[END_REF] Diogenes wonders whether there are honest men.

(13) Diogenes wonders whether there are men.

It is now possible to argue in analogous fashion about (4). ( 14) entails (15). Perhaps (4) does indeed have the Russellian logical form possessed by (8), repeated here as (16). But this does not mean that it is predicted to entail (17). ( 14) There is exactly one ghost in Hans's attic and it will be quiet tonight.

(15) There is exactly one ghost in Hans's attic.

(16) Hans wonders whether there is exactly one ghost in his attic and it will be quiet tonight.

(17) Hans wonders whether there is exactly one ghost in his attic.

This seems to deal with the worry expressed by HKE about (4), and a similar argument could be made about (3).

A First Response to KN's Response

What can be said in response to KN's argument? The first thing to note is that KN do not deal with examples involving conditionals, like (5). Could their argument be extended to deal with such examples? It might seem as if it could. After all, (18) does not entail (19).

(18) If there are honest men, Diogenes will be surprised.

(19) If there are men, Diogenes will be surprised. KN could claim, then, that (9), repeated here as (20), should not be expected to entail (21).

(20) If there is exactly one ghost in his attic and it is quiet tonight, Hans will hold a party.

(21) If there is exactly one ghost in his attic, Hans will hold a party.

Thus KN would be able to argue against all the kinds of example cited by HKE.

I do not, however, believe that KN's argument can profitably be extended along the lines just indicated. For the problem with examples like ( 5) is not that ( 5) is predicted by Russell to entail 4 (21). ( 21) entirely omits the requirement that the ghost in question be quiet tonight, and is thus not remotely plausible as an entailment of ( 5). The problem with (5) and its Russellian paraphrase ( 20) is that we have a strong intuition that the speaker has to be assuming that there is exactly one ghost in Hans's attic in saying (5), whereas no commitment along these lines is made by (20). We can emphasize the difference by preceding the two sentences with a claim not to know the truth of a relevant proposition:

(22) I do not know whether there are any ghosts in Hans's attic. But if the ghost in his attic is quiet tonight, he will hold a party.

(23) I do not know whether there are any ghosts in Hans's attic. But if there is exactly one ghost in his attic and it is quiet tonight, he will hold a party.

Native speakers of English consistently maintain that the speaker of ( 22) sounds self-contradictory.

But the speaker of (23) does not. So the contributions to the truth conditions made in the one place where ( 22) and ( 23) differ cannot be the same. But the clauses in the one place where ( 22) and ( 23) differ are just a sentence involving a definite description and its Russellian paraphrase.

So the Russellian paraphrase cannot be accurate.

Could we salvage the Russellian analysis here by stipulating somehow that definite descriptions take obligatory wide scope over conditionals? Then we would obtain (24) as a paraphrase of ( 22):

(24) I do not know whether there are any ghosts in Hans's attic. But there is an entity such that it is a ghost in Hans's attic and nothing else is a ghost in Hans's attic and if it is quiet tonight Hans will hold a party. This paraphrase would obviously account for the intuition that the speaker of ( 22) is being inconsistent.

It can be demonstrated, however, that definite descriptions do not take obligatory wide scope over conditionals. Let us first remind ourselves of the fact, well established in empirical work on scope islands, 2 that quantifier phrases formed with every cannot take scope out of the antecedents of conditionals. Note that (25) cannot have the reading in ( 26):

(25) If every boy wins a prize, the headmaster will be happy.

(26) Every boy x is such that if x wins a prize the headmaster will be happy.

And ( 27) cannot have the reading in ( 28):

(27) If the headmaster praises every boy, he will be happy.

(28) Every boy x is such that, if the headmaster praises x, x will be happy. Now consider example (29) and the conceivable reading in (30):

(29) If every boy wins the prize he wants, the headmaster will be happy.

(30) Every boy x is such that, if x wins the prize x wants, the headmaster will be happy.

(29) does not have the reading in (30). So every boy has not taken scope over the conditional but takes scope within the antecedent. 3 But this sentence has a very natural reading in which every boy binds he. So he is within the scope of every boy. It is also an integral part of the definite description the prize he wants: he wants is an intersective modifier of prize, and there is no evident way of making it scope higher than prize. So the definite description is within the scope of every boy and is thus within the scope of the conditional. So definite descriptions do not take obligatory wide scope over conditionals. Since this is the case, to resume the main thread of the argument, the Russellian analysis of definite descriptions predicts that (22) will have a reading equivalent to (23). But it does not. 3 I discount the possibility that every boy might take scope over the antecedent of the conditional but not over the whole conditional (antecedent plus nuclear scope). I do not see what this could possibly amount to in semantic terms. And besides, the headmaster's happiness in the current example is clearly contingent on every boy winning the prize he wants; every boy belongs semantically inside the antecedent, then.

A Second Response to KN's Response

I maintain, then, that examples like (5) still constitute a problem for the Russellian analysis of definite descriptions. But have KN relieved the analysis of its apparent embarrassment over examples like (3) and (4)? I believe not.

Suppose Hans sincerely tells us (31). If he then goes on to say (32a), is he being consistent?

And if he goes on to say (32b)?

(31) I am unsure whether there is a ghost in my attic.

(32) a. I am wondering whether there is an entity such that it is a ghost in my attic and nothing else is a ghost in my attic and it is being noisy.

b. I am wondering whether the ghost in my attic is being noisy.

Suppose, again, that Hans sincerely tells us (31). If he then goes on to say (33a), is he being consistent? And if he goes on to say (33b)?

(33) a. I would like there to be an entity such that it is a ghost in my attic and nothing else is a ghost in my attic and it is quiet tonight.

b. I would like the ghost in my attic to be quiet tonight.

Native speakers judge that Hans's propositional attitudes are consistent if he continues with the (a) sentences above, and inconsistent if he continues with the (b) sentences.

HKE's claim, then, should not be that (32a) entails that Hans is wondering whether there is exactly one ghost in his attic, whereas (32b) does not; it should rather be the superficially similar but ultimately distinct claim that the utterance of (32a) is consistent with Hans being unsure whether there is a ghost in his attic, whereas the utterance of (32b) is not. 4 But (32a) is just a Russellian paraphrase of (32b). So the Russellian paraphrase cannot be accurate. An analogous argument can be made with respect to (33).

Once more there is a temptation to try to rescue Russell's analysis here by claiming that definite descriptions must take wide scope over relevant operators-attitude verbs, in this case.

Then, instead of (32a) and (33a), we would have to have ( 34) and ( 35) as paraphrases of (32b) and (33b):

(34) There is an entity such that it is a ghost in my attic and nothing else is a ghost in my attic and I am wondering whether it is being noisy.

(35) There is an entity such that it is a ghost in my attic and nothing else is a ghost in my attic and I would like it to be quiet tonight.

If these were the only meanings available to (32b) and (33b), we would understand why they seemed incompatible with (31). But this analysis is impossible to sustain. Consider ( 36):

(36) Ponce de León is wondering whether the fountain of youth is in Florida.

In an example like (36), assuming that the speaker does not believe in the existence of a fountain of youth, it is claimed as a virtue of the Russellian analysis that definite descriptions are able to take narrow scope with respect to propositional attitude verbs (Neale 1990: 27). For the reading with narrow scope for the definite description would be as follows:

(37) Ponce de León is wondering whether there is an entity such that it is a fountain of youth and nothing else is a fountain of youth and it is in Florida.

This paraphrase, whatever may be its other merits or demerits, at least does not commit the speaker to the existence of a fountain of youth. Contrast the reading in which the definite description takes wide scope over the attitude verb:

(38) There is an entity such that it is a fountain of youth and nothing else is a fountain of youth and Ponce de León is wondering whether it is in Florida.

This clearly commits the speaker to the existence of a fountain of youth. Since (36) does not necessarily involve such a commitment, it is important that the narrow scope reading in (37) is available. But then the Russellian is forced to admit that definite descriptions can take narrow scope with respect to propositional attitude verbs. So the Russellian analysis predicts incorrectly that (32b) and (33b) will have readings equivalent to (32a) and (33a).

The Fregean Analysis of Definite Descriptions

The Fregean analysis of definite descriptions differs from the Russellian approach in that it makes the existence and uniqueness of the ghost presuppositions, not assertions. 5 It seems perfectly well equipped to deal with the examples we have been looking at.

Take (5), repeated here as (39).

(39) If the ghost in his attic is quiet tonight, Hans will hold a party.

We observed earlier that the speaker of this sentence seems to be assuming the existence of a ghost in Hans's attic. We could equally well have said that the speaker is presupposing the existence of such a ghost. On the Fregean analysis of the definite article, the definite description introduces such a presupposition and it becomes a presupposition of the entire sentence. It is well established in work on the 'projection' of presuppositions that the antecedents of conditionals allow presuppositions 5 A number of points should be clarified. Firstly, by 'Fregean analysis' I do not mean an analysis that adopts

Frege's (or Strawson's) view of presupposition. The Fregean analysis of the definite article is in principle neutral between different analyses of presupposition. Whatever kind of thing presuppositions turn out to be, the Fregean analysis can say that the existence and uniqueness entailments associated with definite descriptions are that kind of thing. Secondly, the Fregean analysis is not committed to having definite descriptions be directly referential. They could be individual concepts, for example [START_REF] Elbourne | Situations and Individuals[END_REF][START_REF] Elbourne | Demonstratives as individual concepts[END_REF]. And thirdly, an anonymous reviewer maintains that no current theory of presupposition correctly predicts that the speaker of ( 22) and similar examples will sound inconsistent. But I take this observation, if correct, to have implications for the mechanisms afforded by current theories for dealing with presupposition failure; I do not take it to have implications for the Fregean analysis of descriptions per se.

introduced in them to become presuppositions of the conditional as a whole. 6 The behaviour of (39) follows immediately from the supposition that it (or a speaker who uses it) presupposes the existence of a ghost in Hans attic, as the reader can verify. In particular, it is now obvious why a speaker of ( 22) should sound self-contradictory.

Let us move on to (3) and ( 4), repeated here as ( 40) and ( 41).

(40) Hans wants the ghost in his attic to be quiet tonight.

(41) Hans wonders whether the ghost in his attic will be quiet tonight.

Since these examples involve propositional attitude verbs, we need to know what happens to presuppositions that are introduced by sentences in the scope of such verbs. This question has already been insightfully investigated. [START_REF] Karttunen | Presupposition and linguistic context[END_REF] has observed that in such cases the presuppositions of the embedded sentence come to form part of a new presupposition carried by the whole sentence, namely that the subject of the propositional attitude verb believes the presuppositions of the embedded sentence. For example, it is commonly assumed that (42) carries the presupposition in ( 43):

(42) John has stopped drinking.

(43) John drank.

According to Karttunen, then, when we embed (42) under a propositional attitude verb, as in (44), its presupposition will no longer be present as such; there will instead be a presupposition to the effect that the subject of the attitude verb believes (43).

(44) Mary believes that John has stopped drinking.

This seems to be correct. The proposition that John drank is certainly not a presupposition of (44), since we can felicitously continue as in ( 45).

(45) Mary believes that John has stopped drinking. But in fact John never drank.

And it seems infelicitous now to add that Mary does not believe that John drank:

(46) Mary believes that John has stopped drinking. But she does not believe he drank.

It seems, then, that (44) presupposes that Mary believes that John drank. So Karttunen is correct:

the presupposition of the embedded sentence has been taken and made into a component of the presupposition of the sentence as a whole.7

Following Karttunen, then, we can postulate that the presupposition that there is exactly one ghost in Hans's attic, carried by the embedded sentence in ( 40) and ( 41), is manipulated by the propositional attitude verb and contributes to a presupposition carried by the whole sentence to the effect that Hans believes that there is exactly one ghost in his attic. This, again, seems to be in accordance with our intuitions. In particular, it is now understandable why we attribute inconsistent propositional attitudes to Hans when he follows (31) with (32b) or (33b): he asserts agnosticism with respect to the question of whether there is a ghost in his attic and then presupposes that he believes there is such a ghost.

Conclusion

I submit, then, that the Fregean analysis of definite descriptions is superior to the Russellian analysis with respect to the data examined in this article. This conclusion should be viewed in the context of other recent works arguing in favor of the Fregean analysis.8

2

  The notion of islands for overt syntactic movement goes back to Ross 1967. A classic reference dealing extensively with islands for scope is May 1985. The topic has been dealt with extensively in generative linguistics. For a useful survey, see[START_REF] Szabolcsi | The syntax of scope[END_REF]

It should be noted, however, thatKaplan (2005: 985) expresses a nuanced position that does allow that there might be related problems.

An anonymous reviewer criticized this article on the grounds that wonder and other attitude verbs are not closed under classical consequence; but I hope that this paragraph makes clear that my current criticism of Russell is not based on any assumption to the contrary. My own analysis of these data can be found in the next section.

Antecedents of conditionals are 'holes' for presupposition projection, in[START_REF] Karttunen | Presuppositions of compound sentences[END_REF] terminology. For additional data and a survey of rival theories of presupposition projection, see[START_REF] Beaver | Presupposition[END_REF]

See Heim 1992 for further discussion.

See von Fintel 2004[START_REF] Elbourne | Situations and Individuals[END_REF][START_REF] Glanzberg | Definite descriptions and quantifier scope: some Mates cases reconsidered[END_REF][START_REF] Rothschild | Presuppositions and scope[END_REF]

Acknowledgements I am grateful to David Adger, Ray Buchanan, Paul Dekker, Michael Glanzberg, Gary Ostertag and three anonymous reviewers for comments on various drafts of this article. Previous versions of it were presented as talks at the 1st Lisbon Workshop on Semantics (Peter of Spain Philosophy Institute, University of Lisbon) and at the Philosophy Faculty at Oxford University. I am grateful to the audiences on those occasions for their comments, especially to Delia Graff-Fara, Jeff King, Ofra Magidor, Jason Stanley, Scott Sturgeon, Zoltan Szabó and Tim


Anthony M Grosso
email: anthony.grosso@uclh.nhs.uk

Pritesh N Bodalia

Raymond J Macallister

Aroon D Hingorani

James C Moon

Michael A Scott

Candesartan and Losartan in the Management of Hypertension and Heart Failure: A Systematic Review, Meta-and Cost-Utility Analysis

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Cardiovascular disease (CVD) is the leading cause of morbidity and mortality worldwide, [START_REF] Lopez | Global and regional burden of disease and risk factors, 2001: systematic analysis of population health data[END_REF] despite advances in diagnosis and management. Hypertension is a major risk factor for CVD, and a risk factor for the development of heart failure. The goal of hypertension therapy is to reduce blood pressure (BP) to less than 140/90 mmHg in elderly patients and to less than 130/85 mmHg in the young or middle-aged and in patients with diabetes mellitus irrespective of age. [START_REF] Mansia | ESH-ESC Guidelines for the management of arterial hypertension: the task force for the management of arterial hypertension of the European Society of Hypertension (ESH) and of the European Society of Cardiology (ESC)[END_REF] Patients with symptomatic chronic heart failure (CHF) have reduced cardiac output, characteristic symptoms of dyspnoea, orthopnoea and decreased exercise capacity and have a high risk of death and hospitalisation.

3 Although angiotensin-converting-enzyme (ACE) inhibitors are a well established class of treatment, some patients are unable to tolerate them. In recent years, angiotensin-II receptor blockers (ARBs) have emerged as an alternative option for targeting and inhibiting the renin-angiotensin-aldosterone system (RAAS) by selectively blocking the AT 1 subtype. [START_REF] Timmermans | Angiotensin II receptors and angiotensin II receptor antagonists[END_REF] ARBs exert a similar antihypertensive effect to ACE inhibitors, 5;6 however, their specificity avoids major ACE inhibitor-related adverse effects, such as cough and angioedema, which is believed to result from non-specific interference of bradykinin metabolism. [START_REF] Lacourcie're | Effects of modulators of the renin-angiotensin-aldosterone system on cough[END_REF] Compared with losartan (Cozaar; MSD), the first ARB to receive a marketing authorisation for the management of hypertension, candesartan (Amias; Takeda), the current market leader, has a slower dissociation rate from the AT 1 receptor, 8 potentially providing it with a longer-acting antihypertensive effect.

Until recently both agents were comparably priced, however, the patent for losartan has now expired and cheaper generic alternatives are emerging. We therefore undertook a systematic review and meta-analysis of all published studies comparing losartan and candesartan in a head-to-head randomised controlled trial design to obtain estimates on their relative efficacy and cost-effectiveness. ). The search terms and limits are provided as supplementary material. In addition to the database search strategy, we hand-searched the reference list of identified manuscripts to identify additional relevant studies. To formulate and ensure optimal reporting of the systematic review and meta-analysis, the established PRISMA [Preferred Reporting Items for Systematic reviews and Meta-analysis] statement was followed.

F

9

Inclusion criteria

For hypertension, we included trials if they were of randomised, double-blind, active-controlled design (investigating candesartan and losartan), provided they recruited adult patients (>18 years), were of parallel or cross-over design and the treatment period was of at least four weeks duration. For heart failure, we included trials if they were of randomised, double-blind, active-or placebo-controlled design of adult patients (> 18 years) with a left ventricular ejection fraction (LVEF) not exceeding 40%.

Exclusion criteria

Studies were excluded if they were open-label, observational, or not fully published (e.g. only presented at conference proceedings or solely available in abstract form). Non-English language publications were also excluded.

Endpoints

The primary efficacy endpoints for extraction from the hypertension trials were the mean change from baseline in trough (24 hours post dose) diastolic BP (DBP) and systolic BP (SBP). The primary efficacy endpoint for extraction from the heart failure trials was a composite of cardiovascular death and hospital admission for management of heart failure.

Data from systematic reviews or previous meta-analyses were not used to enable the collection of data from original sources; however any such publications identified served as a comparator to ensure that all relevant studies had been included within this review. Secondary searches were conducted from the reference lists of manuscripts identified. To minimise bias, two investigators (P.N.B. and A.M.G.) reviewed references and abstracts retrieved by the search and selected potentially relevant publications against the pre-specified inclusion and exclusion criteria. Important clinical and methodological study characteristics were extracted onto a standard form, checked, and recorded. Any discrepancies or lack of agreement between the two reviewers were referred to a third independent investigator (A.D.H.) for arbitration. An assessment of risk of bias (using established criteria) [START_REF] Higgins | Cochrane Handbook for systematic reviews of interventions[END_REF] was also undertaken. All analyses were based on intention-to-treat data. For any trials that reported data using a per-protocol analysis, intention-totreat values were extracted.

We used a random-effects model 11;12 to compare outcomes, reported as the weighted mean difference; therefore estimates meta-analysed over multiple trials are average treatment effects. To assess the presence of heterogeneity across pooled studies, the Cochran Q and Higgins I-squared statistics were calculated. [START_REF] Higgins | Measuring inconsistency in meta-analyses[END_REF] Analyses were conducted using StatsDirect® version 2.7.7 (Altrincham, Cheshire, UK). Cost-utility analysis A ten-year cost-utility analysis of adopting losartan or candesartan using a Markov state transition model (see Figure 2) was developed using Microsoft Excel (Microsoft Corporation, 2007). Structurally, the model assumes that all patients can be in one unique health state in a given cycle (one of: 'Well', 'Coronary Heart Disease (CHD)', 'Stroke' and 'Death') with the entire cohort initially in the 'Well' state. We use a ten-year time horizon and annual cycle length for comparison of the costs and health outcomes for patients with essential hypertension, from the viewpoint of the UK National Health Service (NHS). We consider development of primary disease and subsequent quality-adjusted survival.

Baseline risk parameters (age, gender, SBP), smoking status, presence/absence of diabetes or atrial fibrillation (AF) and high density lipoprotein (HDL) levels and drug treatment effects on SBP were computed. A risk sub-model was then used to calculate the age-and sex-related probabilities of stroke and CHD risk for each year in the model, based on Framingham risk equations. 14;15 As noted by Wolf PA et al (1991), SBP is an accurate predictor of stroke risk.

Although, other studies have tried to improve the predictive value of outcomes beyond that which the well established Framingham risk score could predict, a recent review showed that the studies were hampered by methodological flaws. [START_REF] Tzoulaki | Assessment of claims of improved prediction beyond the Framingham risk score[END_REF] We model male and female cohorts separately into baseline risk by SBP level (mild 140 mmHg, moderate 165mmHg, high 180mmHg) and calculate subsequent mortality due to myocardial infarction (MI) and cerebrovascular events.

To overcome the normal fixed temporal probability limitation of standard Markov models, annual time-dependent transition probabilities were calculated using look-up tables for age and gender-related all-cause mortality from UK life tables and data from the MI/stroke mortality in the Framingham follow-up study. [START_REF]Framingham Heart Study, 30-year follow-up[END_REF] Mortality was limited to 10 years post-disease since the regression equations for stroke progression are valid only for a 10-year period. Half-cycle correction was used when calculating life-years. 18 For a one thousand essential hypertension patient cohort simulation, given varying baseline risks of developing CHD and CVD, we calculated the following secondary economic outcomes: total and average costs per strategy, life-years gained, quality-adjusted life-years (QALYs) gained and report the primary outcome of incremental cost-effectiveness ratios (ICERs) of the two treatments by calculating the ratio of the averaged incremental costs to incremental QALYs. An ICER value of £30,000 per QALY was used as an upper threshold for NHS cost-effectiveness.

Best estimates of disease-state costs and utility values for the base-case model were estimated from the published literature. We considered the direct costs for drug acquisition and those associated with managing initial non-fatal stroke and CHD events as well as the costs for ongoing management, inflated from 2005 to 2009 base year values using the Hospital and Community Pay & Price Index. [START_REF] Curtis | Unit Cost of Health and Social Care[END_REF] In line with the National Institute for Health and Clinical Excellence (NICE) Technology Appraisals, annual disease-state quality-of-life was estimated, assuming that MI survivors had higher utility after their initial events and stroke survivors had constant utility thereafter. Cost and health benefits were discounted at a rate of 3.5% in concordance with NICE Technology Appraisal guidance. We assume equal utility loss in both arms due to adverse treatment effects since we are comparing two drugs within the same class without proven evidence of any tolerability differences. The death state is associated with zero cost and utility.

RESULTS

Hypertension

Overall, nine studies fulfilled inclusion criteria for hypertension representing 3090 patients. [START_REF] Andersson | The antihypertensive effect and tolerability of candesartan cilexetil, a new generation angiotensin II antagonist, in comparison with losartan[END_REF][START_REF] Baguet | A placebo-controlled comparison of the efficacy and tolerability of candesartan cilexetil, 8 mg, and losartan, 50 mg, as monotherapy in patients with essential hypertension, using 36-h ambulatory blood pressure monitoring[END_REF][START_REF] Bakris | Antihypertensive efficacy of candesartan in comparison to losartan: the CLAIM study[END_REF][START_REF] Gradman | Comparative effects of candesartan cilexetil and losartan in patients with systemic hypertension. Candesartan Versus Losartan Efficacy Comparison (CANDLE) Study Group[END_REF][START_REF] Vidt | A forced titration study of antihypertensive efficacy of candesartan cilexetil in comparison to losartan: CLAIM Study II[END_REF][START_REF] Lacourciere | A comparison of the efficacy and duration of action of candesartan cilexetil and losartan as assessed by clinic and ambulatory blood pressure after a missed dose, in truly hypertensive patients: a placebo-controlled, forced titration study. Candesartan/Losartan study investigators[END_REF][START_REF] Manolis | Effects of losartan and candesartan monotherapy and losartan/hydrochlorothiazide combination therapy in patients with mild to moderate hypertension. Losartan Trial Investigators[END_REF][START_REF] Willemsen | Disparate systemic and renal blocking properties of angiotensin II antagonists during exogenous angiotensin II administration: implications for treatment[END_REF] A summary of the characteristics of the studies included within this review are given in Table 1. Further details of the studies are included as supplementary material. Of the nine studies, two compared low-dose losartan with low-dose candesartan (50mg versus 8mg); two compared low-dose losartan with mid-dose candesartan (50mg versus 16mg); two compared high-dose losartan with mid-dose candesartan (100mg versus 16mg); and three compared high-dose losartan with high-dose candesartan (100mg versus 32mg). The primary efficacy endpoint data were extracted from these studies and pooled to estimate the weighted average reduction in DBP and SBP from baseline in the two treatment groups (see Figure 3a and Figure 3b respectively). These analyses estimate a between-treatment difference of -1.89 mmHg (95% confidence interval (CI) -2.29 to -1.48) for trough DBP and -2.96 mmHg (95% CI -3.60 to -2.32) for trough SBP in favour of candesartan. Overall, the nine studies generated an I-squared statistic of 32.6% for trough DBP and 52.4% for trough SBP indicating that although the results are statistically significant, there is a mild-to-moderate degree of heterogeneity between the individual studies when combined.

For the purpose of the cost utility-analysis we re-performed the meta-analysis using only the three studies which investigated both ARBs at their respective maximal licensed doses to derive a point estimate based on data reporting on comparative doses (see Figure 4a and Figure 4b for reduction in DBP and SBP from baseline, respectively). This produced a between-treatment difference of -1.96 mmHg (95% CI -2.40 to -1.51) for trough DBP and -3.00 mmHg (95% CI -3.79 to -2.22) for trough SBP in favour of candesartan.

Base-case cost-utility model

The base-case Markov model uses a generic losartan retail price of £6.47 (100mg; 28-day pack price) compared to the list price for candesartan 32mg (£16.13; 28-day pack price) at a moderate baseline risk (SBP: 165mmHg) in a cohort of males and females aged 65 years. The difference in mean trough SBP between candesartan and losartan was obtained from the meta-analysis (-3.00 mmHg). The estimated ICERs for males and females with 'moderate' risk are £44,930 and £53,804, respectively, demonstrating the cost-effectiveness of losartan relative to candesartan at current generic acquisition costs over a 10-year horizon.

Sensitivity analyses

We also modelled four alternative scenarios using one-way sensitivity analyses from the base-case model:

Variation in baseline risk

The baseline risk was varied by increasing the cohort pre-treatment SBP in the range 140-180mmHg. The costeffectiveness of candesartan decreases as the baseline risk lowers, as would be expected as the value of treatment is less when fewer patients develop disease. The range of ICERs is £41,469 -£52,644 for males and £41,591 -£85,244 for females.

Variation in hypertensive effectiveness

We firstly looked at cost-effectiveness variations from the base-case for males and females at the limits of the 95% CI for the trough SBP identified in the random effects meta-analysis (-2.22 to -3.79 mmHg). The ICERs are £71,049 and £87,015 for a 2.22mmHg SBP reduction in males and females, respectively, and £44,930 and £53,804 for a 3.79mmHg SBP reduction in males and females, respectively. Therefore, within the identified pooled range of statistical uncertainty, losartan remains the most cost-effective treatment strategy.

Projected pricing for losartan

Based on previous price falls following patent loss, the acquisition cost of generic losartan may to drop to around £0.88 per pack. Based on these figures the base-case ICER would further increase to £74,901 and £91,368 for males and females, respectively. Hence candesartan becomes increasingly unfavourable.

Variation in age

Finally, we looked at variation in the cost-effectiveness with changes in the cohort starting age for males and females. The ICER at 35 years was £151,140 and £369,075 for males and females, respectively, decreasing to the base-case values at age 65. Therefore, throughout the wide range of initial ages, losartan remains the cost-effective strategy.

Heart Failure

Overall, six studies were identified for heart failure; three studies reporting on losartan and three studies reporting on candesartan; although no published head-to-head studies were found (see Table 1). This analysis provides a comprehensive overall estimate of the between-treatment difference in efficacy (reduction in BP) for candesartan as compared to losartan. A between-treatment difference of -1.96 mmHg (95% CI -2.40 to -1.51) for trough DBP and -3.00 mmHg (95% CI -3.79 to -2.22) for trough SBP in favour of candesartan was observed based on comparative studies at their maximal licensed doses.

Based on our calculated BP-differential and using a 10-year Markov model, we estimate that the cost per QALY-gained will exceed £40,000 if candesartan was used in preference to generic losartan. Our base-case model used a cohort of patients at moderate CHD/CVD risk (baseline SBP of 165 mmHg) and the results demonstrate cost-effectiveness of losartan with ICERs [for candesartan] of £44,930 and £53,804, for males and females, respectively. Sensitivity analyses suggests that the case for losartan adoption holds across plausible variation in CHD and CVD risk, relative hypertension reducing effect, gender and patient cohort starting age. This case is strengthened further in a situation where the acquisition cost of the generic drug falls further # , as predicted, or if the patient is female, younger or mildly hypertensive. In addition, the population level benefit achieved by reducing BP could be attained by using cheaper low dose combination therapy regimens.

Heart Failure

In the absence of comparative trials directly comparing the efficacy of candesartan versus losartan in CHF, a qualitative analysis of the key studies was undertaken as an alternative to a quantitative meta-analysis.

Candesartan

The CHARM-Alternative study [START_REF] Granger | Effects of candesartan in patients with chronic heart failure and reduced left-ventricular systolic function intolerant to angiotensinconverting-enzyme inhibitors: the CHARM-Alternative trial[END_REF] was a placebo-controlled randomised trial of candesartan in 2,028 patients with a LVEF ≤ 40% who were intolerant of an ACE inhibitor (72% cough, 13% symptomatic hypotension, 12% renal dysfunction). The primary composite endpoint was significantly reduced with candesartan (334/1013) in comparison with placebo (406/1015) [hazard ratio (HR) 0.77, 95% CI 0.67-0.89; p=0.0004]. This corresponds to a relative risk reduction (RRR) of 23% (absolute difference = 7%) and a number needed to treat (NNT) of 14 (i.e. fourteen patients need to be treated for the duration of the study to prevent one patient from dying of a cardiovascular event or being hospitalised for treatment of heart failure). The primary endpoint was powered to a statistically significant level by the reduction in hospital admission for CHF [HR 0.68, 95% CI 0.57 to 0.81; p<0.0001] as the reduction in cardiovascular death was non-significant [HR 0.85, 95% CI 0.71 to 1.02; p=0.072].

The CHARM-Added study [START_REF] Weir | Efficacy and tolerability of adding an angiotensin receptor blocker in patients with heart failure already receiving an angiotensin-converting inhibitor plus aldosterone antagonist, with or without a beta blocker. Findings from the Candesartan in Heart failure: Assessment of Reduction in Mortality and morbidity (CHARM)-Added trial[END_REF] (ACE inhibitor + candesartan versus ACE inhibitor + placebo) recruited 2,548 patients with a LVEF ≤ 40% who were receiving an optimal tolerated dose of an ACE inhibitor. The primary composite endpoint (cardiovascular death and hospital admission for HF) was significantly reduced [by 15%] with candesartan (483/1276) in comparison with placebo (538/1272) [Hazard ratio (HR) 0.85, 95% CI 0.75-0.96, p=0.011]. This corresponds to a RRR of 16% (absolute difference = 4.4%) and a NNT of 23. A statistically significant reduction was observed for both components of the primary endpoint: cardiovascular death [HR 0.84, 95% CI 0.72 to 0.98; p=0.029] and hospital admission for CHF [HR 0.83, 95% CI 0.71 to 0.96; p=0.014]. The composite secondary endpoint of all-cause mortality or first CHF hospitalisation was also significantly reduced in the candesartan group [HR 0.87, 95% CI 0.78-0.98; p=0.021]. Of note, at baseline, 55% of patients randomised to receive dual therapy were taking a beta-blocker, whilst only 17% were also taking spironolactone [aldosterone antagonist]. With regards to safety, of the 74 patients treated with candesartan + ACE inhibitor who were also taking spironolactone, three (4%) developed serum potassium levels >6mmol/L compared with one of 71 (1%) in the placebo group (number-needed-to-harm = 33).

Losartan

In comparison, the recently published HEAAL study [START_REF] Konstam | Effects of high-dose versus low-dose losartan on clinical outcomes in patients with heart failure (HEAAL study): a randomised, doubleblind trial[END_REF] (low dose losartan [50mg daily] versus high dose losartan [150mg daily]) recruited 3,846 patients with a LVEF ≤ 40% and a documented intolerance to an ACE inhibitor. The proportion of patients who met the primary composite endpoint (death and admission for HF) was 43% (losartan 150mg) versus 46% (losartan 50mg), which was regarded as a modest yet significant benefit (HR 0.90, 95% confidence intervals 0.82 to 0.99; p=0.027). In comparison with the CHARM-Added study, at baseline, 72% of patients randomised to both arms were taking a beta-blocker, and 38% were also taking an aldosterone antagonist. Overall, the authors reported superiority of losartan 150mg once daily over 50mg once daily for the treatment of CHF. With regards to safety, losartan 150mg daily compared with 50mg daily was noted to cause a significant reduction in GFR (6.1mL/min versus 1.9mL/min; p<0.001, respectively). The incidence of premature discontinuation from therapy as a result of hyperkalaemia, hypotension, renal impairment, and angioedema was non-significantly different between the two arms (p=0.20, p=0.65, p=0.22, and p=0.12, respectively). The investigators put forward the hypothesis that up-titrating

UK Licensing

Both losartan and candesartan hold a marketing authorisation for the treatment of essential hypertension and CHF (in patient with a LVEF ≤ 40%). Two key differences in their licensing criteria are noted: [1] Essential hypertension: candesartan is indicated for adults only, whereas losartan is also indicated for children and adolescents aged 6-18 years; and [2] CHF: candesartan is indicated for use as monotherapy or in combination with an ACE inhibitor (following disease progression, incompatibility, or contraindication), whereas losartan is only indicated as monotherapy in place of an ACE inhibitor.

Strengths & Limitations

The use of meta-analyses in estimating comparative-effectiveness has several strengths but some important limitations are apparent. First, all data included within this analysis were extracted from robust trials which met strict inclusion criteria. The requirement of a randomised, double-blinded, controlled trial ensured that the final dataset would be subject to the least possible amount of bias. Second, the efficacy outcome selected (reduction in SBP) is well accepted as an informative outcome measure concerning hypertension and risk factor for the development of CVD. Third, all included study manuscripts were published in full allowing for intention-to-treat analyses.

However, the results obtained from our meta-analysis are subject to the limitations that are inherent in any metaanalysis. First, individual prospective studies only provide information over a short period of time (4 to 12 weeks) whilst the implications are extrapolated for life-long therapy. Second, excluding trials that have not been published may exaggerate the treatment effects observed as publications tend to favour those with positive results. Third, pooling of data from trials with differences in trial design, methodology, and patient groups may result in a heterogeneous dataset from which conclusions are drawn. However, such differences in patient groups may serve to strengthen the meta-analysis by allowing generalisability of the results to a broader group. Furthermore, the use of a random-effects model and tests to identify the presence of significant heterogeneity aid to minimise and highlight the impact of such effect. Fourth, this analysis was restricted to data relating to the adult population, therefore it cannot be directly extrapolated to patients under the age of 18 years. Fifth, doses employed within the clinical trials are not always consistent with those used in clinical practice, therefore limiting the external validity of these data. To minimise the extent of this, only data comparing high-dose losartan with high-dose candesartan were used in the subsequent cost-utility analysis. Finally, data within this analysis only relates to the use of losartan and candesartan for the treatment of hypertension in patients with no other co-morbidities and therefore cannot be directly extrapolated to all patients.

We used best evidence and the well validated Framingham risk equations as criteria for the development of disease and subsequent mortality in our cost-utility analysis. Our model is however subject to some caveats. The model calculates only the first episode of MI or stroke events and the subsequent quality-of-life and costs associated with survivors. It does not consider that stroke patients may experience a fatal MI or vice versa. Whilst we could have constructed a fully probabilistic model, for ease of understanding and presentation we used deterministic analyses allowing for uncertainty using sensitivity analysis. This showed, that under plausible variation in key parameter values using one-way sensitivity analyses, that losartan remains the more cost-effective of the two studied ARBs.

Place in therapy for ARBS Currently there are 11 ACE inhibitors and seven ARBs available in the UK. 31 The UK National Institute for Health and Clinical Excellence (NICE) offers guidance on the use of these drugs in the following areas: CHF, hypertension, MI (secondary prevention), type II diabetes, and chronic kidney disease. Where either an ACE inhibitor or an ARB is indicated, NICE recommend that an ACE inhibitor is routinely the drug of choice on the basis that there is a more robust evidence-base for their use. A recent review also suggested that ARBs may be less effective than even cheaper ACE inhibitors for MI protection. [START_REF] Staessen | Implications of Recently Published Trials of Blood Pressure-Lowering Drugs in Hypertensive or High-Risk Patients[END_REF] However a meta-analysis has suggested that ARBs are indeed as effective as ACE inhibitors on the risk of MI, cardiovascular mortality and total mortality and also concluded that they may even be slightly more protective than ACE inhibitors on the risk of stroke. [START_REF] Reboldi | Comparison between angiotensinconverting enzyme inhibitors and angiotensin receptor blockers on the risk of myocardial infarction, stroke and death: a meta-analysis[END_REF] As CHD is more common than cerebrovascular disease it seems reasonable that ACE inhibitors remain first-line in all (except perhaps in some high-risk groups).

For the treatment of hypertension, if either an ACE inhibitor or an ARB is indicated, the NICE recommendation stipulates using a drug that can be taken once daily, is generically prescribed, and minimises cost. [START_REF]Hypertension: management of hypertension in adults in primary care[END_REF] For heart failure, combined treatment with both an ACE inhibitor and an ARB currently has a limited role. For patients with heart failure who remain symptomatic despite the use of a diuretic, beta-blocker, and ACE inhibitor, options include to either add in and aldosterone antagonist, an ARB or hydralazine combined with a nitrate. 35 Routine treatment with an ACE inhibitor in combination with an ARB is not recommended as recent publications have found little evidence to support this. 36;37 A recently published meta-analysis regarding the use of an ARB in combination with an ACE inhibitor concluded that there was no clear survival benefits associated with the combination treatment strategy.

38

ARB prescribing recommendations

We recommend that generic losartan be initiated in all patients indicated for an ARB in both hypertension and heart failure. ARBs should not routinely be combined with ACE-inhibitors and primarily be used in patients who are ACEinhibitor intolerant. For example, for existing patients on candesartan for hypertension we recommend changing to losartan except in the rare scenario of prior intolerance to losartan. For existing patients on candesartan for heart failure where not already on maximal target dose, ARB dose escalation is encouraged. If a patient on candesartan is due for dose escalation, we recommend changing to losartan at this point unless intolerant. Where a patient is on maximal dose candesartan for heart failure, we recommend that the decision to switch be considered on a case-bycase basis by the responsible physician.

Dosing

The losartan dose used in hypertension should be 50mg or 100mg daily depending on whether the existing ARB dose is at the lower or upper end of its dosing schedule. 150mg daily is now the target dose for heart failure [unlicensed].

CONCLUSION

Although candesartan, the most widely prescribed ARB, appears to reduce BP to a slightly greater extent when compared to losartan, such difference is of questionable clinical significance in the context of combination drug regimens and does not appear to be cost-effective based on current and future acquisition costs of losartan and perceived NHS affordability thresholds. We could find no robust evidence supporting the superiority of candesartan over losartan in the treatment of heart failure. We therefore recommend adopting generic losartan as the ARB of choice which could, based on 2009 prescribing figures in primary care alone, save the UK NHS approximately £200 million per annum in drug costs.

# At the time of publication, the price of generic losartan had fallen to £3 per pack.

Competing interests: We declare no support from any organisation for the submitted work; no financial relationships with any organisations that might have an interest in the submitted work in the previous three years and no other relationships or activities that could appear to have influenced the submitted work.

Exclusive licence: The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the publishers. Figure 4b: Forest plot of the weighted mean difference (WMD) for absolute difference in trough systolic blood pressure from baseline to end of study period (4 to 12 weeks) of randomised controlled trials comparing losartan with candesartan at their maximum licensed doses as monotherapy for the treatment of mild-to-moderate hypertension. The black squares represent the WMD for individual studies and the horizontal line represents the 95% confidence interval of the WMD. The diamond and horizontal line within its represents the random-effects pooled WMD and its corresponding 95% confidence interval. A WMD < 0 is indicative of a significant difference in trough systolic blood pressure in favour of candesartan.

  losartan as monotherapy may provide equally favourable results to a combination of ACE inhibitor and ARB as demonstrated in the CHARM-Added study.

FFigure 2 :Figure 3a :

 23a FIGURES

Figure 3b :Figure 4a :

 3b4a Figure 3b: Forest plot of the weighted mean difference (WMD) for absolute difference in trough systolic blood pressure from baseline to end of study period (4 to 12 weeks) of randomised controlled trials comparing losartan with candesartan as monotherapy for the treatment of mild-tomoderate hypertension. The black squares represent the WMD for individual studies and the horizontal line represents the 95% confidence interval of the WMD. The diamond and horizontal line within it represents the random-effects pooled WMD and its corresponding 95% confidence interval. Estimates to the left of the vertical line (i.e. WMD < 0) are indicative of a significant difference in trough systolic blood pressure in favour of candesartan. Statistical significance is inferred where the confidence interval does not cross the vertical line of unity.

Figure 1 :

 1 Figure 1: Incremental cost-effectiveness ratios (ICERs) of candesartan and losartan versus baseline hypertension risk. Results are displayed for males and females at two generic losartan prices: Price 1 (£77.64 p.a.) and Price 2 (£10.56 p.a.).

Figure 2 :

 2 Figure2: One-way sensitivity analysis on the Incremental cost-effectiveness ratio (ICER) of candesartan relative to losartan versus difference in relative anti-hypertensive effect on systolic blood pressure (SBP) of candesartan relative to losartan.

Figure 3 :

 3 Figure 3: One-way sensitivity analysis on the Incremental cost-effectiveness ratio (ICER) of candesartan relative to losartan versus cohort starting age in the age range (35 -65 years) for males and females.

  Primary exclusion criteria• Systematic reviews • Non-randomised trials • Trials which were of less than 4 weeks duration • Trials which permitted the use of other therapies which may have confounded the clarity of the outcome of the drug being assessed, e.g. calcium channel blockers • Trials which used a response conditional design where patients were allocated treatment only if they showed a predetermined response to treatment during a baseline period before randomisation • Trials which recruited patients into the randomised, double-blind phase following an open-label period (singleblind permitted)

  We searched for relevant randomised trials in the Cochrane Central Register of Controlled Trials (Cochrane Library 2009, issue 2), which contains the Hypertension and Heart Group's specialist register, Medline (1950 to February 2010), and Embase (1980 to February 2010

	METHODS
	Systematic review & meta-analysis
	o r
	P
	e
	e r
	R
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	v i e
	w
	O n l
	y

Table 1 :

 1 Characteristics of studies included within the systematic review of candesartan and losartan in the treatment of hypertension and heart failure.

		Effect size meta-analysis plot [random effects]
	Gradman et al (1999) -100mg vs 32mg
	Bakris et al (2001) -100mg vs 32mg
	Vidt et al (2001) -100mg vs 32mg
	-3.0	-2.5	-2.0	-1.5	-1.0	-0.5
	DL pooled weighted mean difference = -1.960107 (95% CI = -2.407149 to -1.513065)
		Effect size meta-analysis plot [random effects]
	Gradman et al (1999) -100mg vs 32mg
	Bakris et al (2001) -100mg vs 32mg
	Vidt et al (2001) -100mg vs 32mg
	-4.1	-3.1	-2.1	-1.1		-0.1 0
	DL pooled weighted mean difference = -3.000486 (95% CI = -3.785843 to -2.215129)

*Weeks of treatment refers to the double-blind period consisting of both the titration-to-target and target-stabilisation phases (where applicable).

Table 2 :

 2 Cost and utility parameters used in the base-case cost-utility model comparing candesartan and losartan for the treatment of hypertension.*Cost inflated to 2009 base year.

	Parameter	Base-case value	Source
	Annual drug cost of candesartan (32mg)	£193.56	BNF (2009) 31
	Annual drug cost of generic losartan (100mg)	£77.64	TEVA pharmaceuticals	43
	Projected annual drug cost of generic losartan	£10.56	Expert opinion
	(100mg)
	Annual cost of stroke survivor [cost in first year]	£2163 [£8046*]	NICE Guideline CG034 34
	Annual cost of MI survivor [cost in first year]	£500 [£4448*]	NICE Guideline CG034 34
	Utility weight of stroke survivor	0.63	NICE Guideline CG034	34
	Utility weight of MI survivor [first year]	0.88 [0.76]	NICE Guideline CG034 34
	Treatment effect difference (incremental reduction in	-3.00mmHg	Meta-analysis
	SBP)

Table 3 :

 3 Variation of ICER with baseline risk when comparing candesartan and losartan for the treatment of hypertension. .

	Baseline Risk / SBP	ICER (£ / QALY)
	Males	Generic losartan (£77.64 p.a.)	Projected generic price (£10.56 p.a.)
	Mild (140mmHg)	£52,644	£87,946
	Moderate (165mmHg)	£44,930	£74,901
	High (180mmHg)	£41,469	£69,076
	Females
	Mild (140mmHg)	£85,244	£142,449
	Moderate (165mmHg)	£53,804	£91,368
	High (180mmHg)	£41,591	£71,430

Table 1 :

 1 Search strategy

	Search no.	Search term
	#1	Losartan [MeSH]
	#2	Candesartan
	#3	Candesartan cilexetil
	#4	Hypertension [MeSH]
	#5	Heart failure [MeSH]
	#6	#1 AND (#2 OR #3) AND #4
	#6	#1 AND (#2 OR #3) AND #5

Table 2 :

 2 Limits applied within each database searched

	Database	Limits applied
	PubMed	Humans; Randomised controlled trial; English; all adult: 19+ years
	Cochrane	Nil
	Embase	Full text; Human; English language ; Article OR Erratum; Adult <18 to 64 years> OR aged <65 years+
	Primary inclusion criteria
	• Adults
	• Randomised controlled trial
	• Double-blind
	• Placebo-or active-controlled
	• Duration greater or equal to 4 weeks
	• Hypertension (primary)
	• Heart failure
	• English language

Table 3 :

 3 Raw data extracted from clinical trials included within meta-analyses

	Drug	Losartan


Alexander Kolpakov

Alexander Mednykh

Marina Paškevič

Volume formula for a Z 2 -symmetric spherical tetrahedron through its edge lengths *

Keywords: tetrahedron, spherical space, volume, Gram matrix

The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle π in the middle points of a certain pair of its skew edges.

Introduction

The volume calculation problem stated for a three-dimensional polyhedra is one of the most hard and old problems in the field of geometry. The first results belonging to the field is due to N. Fontana Tartaglia (1499-1557), who found a formula for the volume of a Euclidean tetrahedron better know in the present time as the Cayley-Menger determinant. Due to the paper [START_REF] Kh | The volume as a metric invariant of polyhedra[END_REF] the volume of every Euclidean polyhedron is a root of an algebraic equation depending on its combinatorial type and metric parameters.

In case of hyperbolic and spherical spaces the task becomes harder. The formulae for volumes of orthoschemes are known since the work of N. Lobachevsky and L. Schläfli. The volumes of hyperbolic polyhedra with at least one ideal vertex and hyperbolic Lambert cube are given in the papers [START_REF] Alekseevskiì | Geometry-2: Geometry of spaces of constant curvature[END_REF][START_REF] Kellerhals | On the volume of hyperbolic polyhedra[END_REF][START_REF] Mednykh | On hyperbolic and spherical volumes for link conemanifolds[END_REF].

The general formula for the volume of a non-euclidean tetrahedron were given in [START_REF] Cho | On the volume formula for hyperbolic tetrahedra[END_REF][START_REF] Murakami | On the volume of hyperbolic and spherical tetrahedron[END_REF] as a linear combination of dilogarithm functions depending on the dihedral angles. Afterwards, an elementary integral formula was suggested in [START_REF] Derevnin | A formula for the volume of a hyperbolic tetrahedron[END_REF].

In case given polyhedron possesses certain symmetries, the volume formulae become facile. First, this fact was noted by Lobachevsky for ideal hyperbolic tetrahedra: the vertices of such tetrahedra belong to the ideal boundary of hyperbolic space and dihedral angles along every pair of skew edges are equal. Later, J. Milnor presented the respective result in very elegant form [START_REF] Milnor | The Schläfli differential equality[END_REF]. The general case of a tetrahedron with the same kind of symmetry is considered in [START_REF] Derevnin | On the volume of a symmetric tetrahedron in hyperbolic and spherical spaces[END_REF]. For more complicated polyhedra one also expects the use of their symmetries to be an effective tool. The volumes of octahedra enjoying certain symmetries were computed in [1].

The volume formula for a hyperbolic tetrahedron in terms of its edge lengths instead of dihedral angles was suggested first by [START_REF] Murakami | A volume formula for hyperbolic tetrahedra in terms of edge lengths[END_REF] in view of the Volume Conjecture due to R. Kashaev [START_REF] Kashaev | The hyperbolic volume of knots from the quantum dilogarithm[END_REF]. The further investigation on the subject was carried out in [START_REF] Luo | 3-dimensional Schläfli formula and its generalization[END_REF].

The paper [START_REF] Murakami | On the volume of hyperbolic and spherical tetrahedron[END_REF] suggests a volume formula for a spherical tetrahedron as an analytic continuation of the given volume function for a hyperbolic one. The corresponding analytical strata has to be chosen in the unique proper manner.

The present papers provides volume formula for a spherical tetrahedron that is invariant up to isometry under rotation through angle π in the middle points of a certain pair of its skew edges. The formula itself depends on the edge lengths of given tetrahedra as well as on its dihedral angles and specifies the actual analytic strata of the volume function. Volumes of spherical Lambert cube and spherical octahedra with various kinds of symmetry were obtained in [1,[START_REF] Derevnin | The Volume of the Lambert Cube in Spherical Space[END_REF]. The analytic formulae for these polyhedra are of simpler form in contrast to their more complicated combinatorial structure.

Preliminary results

Let R n+1 = {x = (x 0 , . . . , x n ) : x i ∈ R, i = 1, . . . , n} be Euclidean space equipped with the standard inner product x, y = n i=0 x i y i and norm x =

x, x . Let p 0 , . . . , p n be vectors in R n+1 . Define a cone over a collection of vectors p 0 , . . . , p n as cone {p 0 , . . . ,

p n } = n i=0 λ i p i : λ i ≥ 0, i = 1, . . . , n .
Call a spherical n-simplex S the intersection of the cone over collection p 0 , . . . , p n of linearly independent unitary vectors and the n-dimensional sphere

S n = {v ∈ R n : v, v = 1}. Thus S = cone {p 0 , . . . , p n } ∩ S n .
Call p 0 , . . . , p n the vertices of a simplex S. Notice that {p 0 , . . . , p n } ⊂ S n . Call a (k -1)-dimensional face of S the intersection of the cone over a k-element sub-collection of linearly independent vectors {p i 1 , . . . ,

p i k } ⊂ {p 0 , . . . , p n } with {i 1 < • • • < i k } ⊂ {0, . . . , n}, 0 ≤ k ≤ n and the sphere S n .
The matrix G ⋆ = { p i , p j } n i,j=0 is the edge matrix of a simplex S. Let M = {m ij } n i,j=0 be a matrix. Denote by M (i, j) the matrix obtained from M by deletion of i-th row and j-th column for i, j = 0, . . . , n. Put

M ij = (-1) i+j det M (i, j). The quantity M ij is the (i, j)-cofactor of M . Then, call cof M = {M ij } n i,j=0
the cofactor matrix of a matrix M . The unit (outer) normal vector v i , i = 0, . . . , n to a (n -1)-dimensional face S i = {p i 1 , . . . , p in } {p i } of the simplex S is defined by (cf. [START_REF] Luo | On a problem of Fenchel[END_REF])

v i = n k=0,k =i G ⋆ ik p k G ⋆ ii det G ⋆ .
The matrix G = { v i , v j } n i,j=0 is the Gram matrix of a simplex S. Given a simplex S ⊂ S n with vertices {p 1 , . . . , p n } and unit (outer) normal vectors {v 1 , . . . , v n } define its edge lengths by cos l ij = p i , p j and (inner) dihedral angles by cos α ij = -v i , v j with 0 ≤ l ij , α ij ≤ π, i, j = 0, . . . , n. Then the Gram matrix of S is G = {-cos α ij } n i,j=0 and the edge matrix of S is G ⋆ = {cos l ij } n i,j=0 . The sphere S n is endowed with the natural metric of constant sectional curvature +1. Call the given metric space the spherical space S n . The isometry group of the spherical space S n is the orthogonal group O(n + 1). Orientation-preserving isometries of S n compose the subgroup of index two in O(n + 1), called SO(n + 1).

The following theorems tackle existence of a spherical simplex with given Gram matrix or edge matrix [START_REF] Luo | On a problem of Fenchel[END_REF]:

Theorem 1 The Gram matrix {-cos α ij } n i,j=0
of a spherical n-simplex is symmetric, positive definite with diagonal entries equal to 1. Conversely, every positive definite symmetric matrix with diagonal entries equal to 1 is the Gram matrix of a spherical n-simplex that is unique up to an isometry.

Theorem 2

The edge matrix {cos l ij } n i,j=0 of a spherical n-simplex is symmetric, positive definite with diagonal entries equal to 1. Conversely, every positive definite symmetric matrix with diagonal entries equal to 1 is the edge matrix of a spherical n-simplex that is unique up to an isometry.

The following theorem due to Ludwig Schläfli relates the volume of a given simplex in the spherical space S n with volumes of its apexes and dihedral angles between its faces [START_REF] Milnor | The Schläfli differential equality[END_REF][START_REF] Schläfli | dx dy . . . dz whose limits are p 1 = a 1 x+b 1 y +• • •+h 1 z[END_REF]:

Theorem 3 (Schläfli formula) Let a simplex S in the space S n , n ≥ 2, of constant sectional curvature +1 have dihedral angles α ij = ∠ S i S j , 0 ≤ i < j ≤ n formed by the (n -1)-dimensional faces S i and S j of S which intersect in the (n -2)-dimensional apex S ij = S i ∩ S j .
Then the differential of the volume function Vol n on the set of all simplices in S n satisfies the equality

(n -1) dVol n S = n i<j=0 Vol n-2 S ij dα ij where Vol n-1 S ij is the (n -2)-dimensional volume function on the set of all (n -2)-dimensional apexes S ij , Vol 0 S ij = 1, 0 ≤ i < j ≤ n,
and α ij is the dihedral angle between S i and S j along S ij .

The Schläfli formula for the spherical space S 3 can be reduced to dVol S = 1 2

3 i<j=0 l ij dα ij ,
where Vol = Vol 3 is the volume function, l ij represents the length of ij-th edge and α ij represents the dihedral angle along it. So the volume of a simplex in S 3 is related with its edge lengths and dihedral angles. Given a simplex S ⊂ S n with vertices {p 1 , . . . , p n } and unit normal vectors {v 1 , . . . , v n } denote its dual S ⋆ ⊂ S n as a simplex with vertices {v 1 , . . . , v n } and unit normal vectors {v 1 , . . . , v n }.

In case of the spherical space S 3 every edge p i p j , 0 ≤ i < j ≤ 3 of S corresponds to the edge v 3-j v 3-i of its dual S ⋆ . The theorem below was originally discovered by Italian mathematician Duke Gaetano Sforza and could be found in the book [START_REF] Milnor | The Schläfli differential equality[END_REF].

Theorem 4 Let S be a simplex in the spherical space S 3 and let S ⋆ be its dual. Then

Vol 3 S + Vol 3 S ⋆ + 1 2 E⊂S Vol 1 E Vol 1 E ⋆ = π 2 ,
where the sum is taken over all edges E of S and E ⋆ denotes the edge of S ⋆ corresponding to E.

In what follows we call 3-dimensional simplex a tetrahedron for the sake of brevity.

Trigonometric identities for a spherical tetrahedron

Let T be a tetrahedron in the spherical space S 3 with vertices p 0 , p 1 , p 2 , p 3 , dihedral angles A, B, C, D, E, F and edge lengths l A , l B , l C , l D , l E , l F . In the sequel we have that 0

≤ A, B, C, D, E, F ≤ π and 0 ≤ l A , l B , l C , l D , l E , l F ≤ π.
G ⋆ = {g ⋆ ij } 3 i,j=0 =     1 cos l A cos l B cos l C cos l A 1 cos l F cos l E cos l B cos l F 1 cos l D cos l C cos l E cos l D 1    
the edge matrix of T and by

G = {g ij } 3 i,j=0 =     1 -cos D -cos E -cos F -cos D 1 -cos C -cos B -cos E -cos C 1 -cos A -cos F -cos B -cos A 1     its Gram matrix.
Denote by c ij and c ⋆ ij the respective cofactors of the matrices G and G ⋆ for i, j = 0, 1, 2, 3.

Further, we mention several important trigonometric relations (see, e.g. [START_REF] Derevnin | On the volume of a symmetric tetrahedron in hyperbolic and spherical spaces[END_REF]) to be used below:

Theorem 5 (Sine Rule) Given a spherical tetrahedron T with the Gram matrix G and the edge matrix

G ⋆ , denote ∆ = det G, ∆ ⋆ = det G ⋆ , p = c 00 c 11 c 22 c 33 , p ⋆ = c ⋆ 00 c ⋆ 11 c ⋆ 22 c ⋆ 33 . Then sin l A sin l D sin A sin D = sin l B sin l E sin B sin E = sin l C sin l F sin C sin F = ∆ √ p = √ p ⋆ ∆ ⋆ .
Theorem 6 (Cosine Rule) For the respective pairs of skew edges of a spherical tetrahedron T the following equalities hold:

cos l A cos l D -cos l B cos l E cos A cos D -cos B cos E = cos l B cos l E -cos l C cos l F cos B cos E -cos C cos F = = cos l C cos l F -cos l A cos l D cos C cos F -cos A cos D = ∆ √ p = √ p ⋆ ∆ ⋆ .
Also we need the following theorem due to Jacobi (see [15, Théorème 2.5.2]):

Theorem 7 Let M = {m ij } n i,j=0 be a matrix, cof M = {M ij } n i,j=0 be its cofactor matrix, 0 < k < n and σ = i 0 . . . i n j 0 . . . j n an arbitrary permutation.
Then

det{M ipjq } k p,q=0 = (-1) sgn σ (det M ) k det{m ipjq } n p,q=k .
4 Trigonometric identities for a Z 2 -symmetric spherical tetrahedron

Consider a spherical tetrahedron T which is symmetric under rotation through angle π about the axe that passes through the middle points of the edges p 0 p 1 and p 2 p 3 . We call such tetrahedron Z 2 -symmetric. Note, that in this case

l B = l E , l C = l F and B = E, C = F .
(i) l A = l D if and only if A = D, (ii) l A > l D if and only if A < D.
Proof. As applied to the edge matrix G ⋆ , Theorem 7 grants the equality

c ⋆ 00 c ⋆ 23 -c ⋆ 01 c ⋆ 22 = ∆ ⋆ (g ⋆ 01 -g ⋆ 23 ).
Recall,

g 01 = -cos D = c ⋆ 01 c ⋆ 00 c ⋆ 11 , g 23 = -cos A = c ⋆ 23 c ⋆ 22 c ⋆ 33 , g ⋆ 01 = cos l A , g ⋆ 23 = cos l D . For a spherical Z 2 -symmetric tetrahedron we have c ⋆ 00 = c ⋆ 11 > 0, c ⋆ 22 = c ⋆ 33 > 0. Thus, cos l A -cos l D = - c ⋆ 00 c ⋆ 22 ∆ ⋆ (cos A -cos D).
As far as 0 ≤ l A , l D ≤ π and 0 ≤ A, D ≤ π, the assertions of the Lemma follow.

Trigonometric identities from in Theorem 5 and Theorem 6 imply

Proposition 1 Let T be a Z 2 -symmetric spherical tetrahedron with dihedral angles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D .
Then the following equalities hold:

u = sin l A +l D 2 sin A+D 2 = sin l A -l D 2 sin D-A 2 = sin l B sin B = sin l C sin C = v -1 ,
where

u = c ⋆ 00 c ⋆ 22 ∆ ⋆ , v =
c 00 c 22 ∆ correspond to the principal and the dual parameters of the tetrahedron T.

Proof. Recall the common property of ratios:

a b = c d = a -c b -d = a + c b + d .
By use of the equalities above and trigonometric identities cos(ϕ + ψ) = cos ϕ sin ψ -cos ψ sin ϕ, cos(ϕ -ψ) = cos ϕ sin ψ + cos ψ sin ϕ we deduce from Theorem 5 and Theorem 6 the following relations:

∆ c 00 c 22 = 1 -cos(l A + l D ) 1 -cos(A + D) = 1 -cos(l A -l D ) 1 -cos(D -A) = sin 2 l B sin 2 B = sin 2 l C sin 2 C = c ⋆ 00 c ⋆ 22 ∆ ⋆ . Put u = c ⋆ 00 c ⋆ 22 ∆ ⋆ , v = c 00 c 22
∆ . The quantities u and v are positive real numbers, related together by uv = 1. We call u the principal parameter of the tetrahedron T and v is called its dual parameter. Granted the identity 1 -cos ϕ = 2 sin 2 ϕ 2 it follows that

u 2 = sin 2 l A +l D 2 sin 2 A+D 2 = sin 2 l A -l D 2 sin 2 D-A 2 = sin 2 l B sin 2 B = sin 2 l C sin 2 C = v -2 .
Extracting square roots in accordance with Lemma 1 finishes the proof.

5 Volume formula for a Z 2 -symmetric spherical tetrahedron 5.1 Further trigonometric identities for a Z 2 -symmetric spherical tetrahedron

Let T be a Z 2 -symmetric spherical tetrahedron with dihedral angles A,

B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Denote a + = cos l A + l D 2 , a -= cos l A -l D 2 , b = cos l B , c = cos l C and A + = cos A + D 2 , A -= cos D -A 2 , B = cos B, C = cos C.
The Lemma below gives a useful identity that follows from the definition of the principal parameter u for the tetrahedron T: Lemma 2 The principal parameter u of the tetrahedron T is a positive root of the quadratic equation

u 2 + 4 (a + a --bc)(a + b -a -c)(a + c -a -b) ∆ ⋆ = 1,
where

∆ ⋆ = (a + + a -+ b + c)(a + + a --b -c)(a + -a --b + c)(a + -a -+ b -c).
Proof. Substitute u from Proposition 1, then express the product of c ⋆ 00 and c ⋆ 22 as a polynomial in the new variables a + , a -, b, c and proceed with straightforward computations.

As for the dual parameter of T, the following Lemma holds:

Lemma 3
The dual parameter u of the tetrahedron T is a positive root of the quadratic equation

v 2 - 4 (A + A -+ BC)(A + B + A -C)(A + C + A -B) ∆ = 1,
where

∆ = (A --A --B-C)(A --A + +B+C)(A -+A + -B+C)(A -+A + +B-C).
In what follows we say a spherical tetrahedron T s with dihedral angles α, β, γ, δ, ε, ϕ and edge lengths l α , l β , l γ , l δ , l ε , l ϕ to be symmetric if

l α = l δ , l β = l ε , l γ = l ϕ or, that is equivalent, α = δ, β = ε, δ = ϕ. Denote ã = cos l α , b = cos l β , c = cos l γ , Ã = cos α, B = cos β, C = cos γ,
The following trigonometric identities are proven in [START_REF] Derevnin | On the volume of a symmetric tetrahedron in hyperbolic and spherical spaces[END_REF]:

Proposition 2 Let T s be a symmetric spherical tetrahedron with dihedral angles α = δ, β = ε, γ = ϕ and edge lengths l α = l δ , l β = l ε , l γ = l ϕ . Then the following equalities hold:

sin l α sin α = sin l β sin β = sin l γ sin γ = u s ,
where the positive root of quadratic equation

u 2 s + 4(ã -bc)( b -ãc)(c -ãb ) δ ⋆ = 1 with δ ⋆ = (ã + b + c + 1)(ã -b -c + 1)( b -ã -c + 1)(c -ã -b + 1)
represents the principal parameter u s of the tetrahedron T s .

Meanwhile, the dual parameter of T s is the positive root of the following equation:

v 2 s - 4( Ã + B C)( B + Ã C)( C + Ã B) δ = 1 with δ = (1 -Ã -B -C)(1 -Ã + B + C)(1 + Ã -B + C)(1 + Ã + B -C).
The following lemma shows the correspondence between Z 2 -symmetric and symmetric spherical tetrahedra.

Lemma 4 Let T be a Z 2 -symmetric spherical tetrahedron with dihedral angles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Then there exists the associated symmetric tetrahedron T s with the Gram matrix

G s =     1 -cos α -cos β -cos γ -cos α 1 -cos γ -cos β -cos β -cos γ 1 -cos α -cos γ -cos β -cos α 1     = =       1 -A + A --B A --C A - -A + A - 1 -C A --B A - -B A --C A - 1 -A + A - -C A --B A --A + A - 1      
and the edge matrix

G ⋆ s =     1 cos l α cos l β cos l γ cos l α 1 cos l γ cos l β cos l β cos l γ 1 cos l α cos l γ cos l β cos l γ 1     =      1 a + a - b a - c a - a + a - 1 c a - b a - b a - c a - 1 a + a - c a - b a - a + a - 1      . Proof. Denote the principal cofactors of G ⋆ s by s ⋆ ii , i = 0, 1, 2, 3.
To prove existence of T s it suffices by Theorem 2 to show that det G ⋆ s > 0 and s ⋆ ii > 0, i = 0, 1, 2, 3. We have that det

G ⋆ s = det G ⋆ a 4 - > 0 and s ⋆ 00 - c ⋆ 00 a 2 - = - 2 a 2 - (a + a --bc) sin l A -l D 2 sin l A , s ⋆ 00 - c ⋆ 22 a 2 - = 2 a 2 - (a + a --bc) sin l A -l D 2 sin l D .
From the former two equalities we deduce that depending on the sign of their right-hand parts either s ⋆ 00 ≥

c ⋆ 00 a 2 - or s ⋆ 00 ≥ c ⋆ 22 a 2 -
. As far as the tetrahedron T exists, c ⋆ 00 > 0 and c ⋆ 22 > 0. It follows that s ⋆ 00 = s ⋆ 11 = s ⋆ 22 = s ⋆ 33 > 0. Thus, the tetrahedron T s exists.

For the edge lengths of the symmetric spherical tetrahedron T s one has

cos l α = a + a - , cos l β = b a - , cos l γ = c a - .
Let us prove that

cos α = A + A - , cos β = B A - , cos γ = C A - .
Refer to Proposition 1 together with Lemma 2 and note that the following relation holds between the principal parameters u and u s of tetrahedra T and T s , respectively:

1 -u 2 = a 2 -(1 -u 2 s ). Substituting u = sin l A +l D 2 sin A+D 2
from Proposition 1 and u s = sin lα sin α from Proposition 2 to the relation above one obtains

cos 2 α = cos 2 A+D 2 1 - sin 2 l A -l D 2 u 2 = cos 2 A+D 2 cos 2 D-A 2 = A 2 + A 2 - .
Thus,

cos α = ± cos A+D 2 cos D-A 2 = ± A + A - .
We should choose the proper sign in the equality above. Note, that if T is symmetric, i.e. l A = l D , then G ⋆ s = G ⋆ . That means T has isometric associated symmetric tetrahedron T s . Thus, the Gram matrices for tetrahedra T and T s mentioned in the assertions of the Lemma coincide. For the equality

G s = G to hold if T is symmetric, we put cos α = cos A+D 2 cos D-A 2 = A + A - .
The rest of the proof follows by analogy. Denote the auxiliary parameters of the tetrahedron T:

t 2 = 1 -u 2 = 4 (a + a --bc)(a + b -a -c)(a + c -a -b) ∆ ⋆ and τ 2 = v 2 -1 = 4 (A + A -+ BC)(A + B + A -C)(A + C + A -B) ∆ .
By Lemma 2 the quantity t could be either real or pure imaginary. We choose t to be non-negative or to have non-negative imaginary part. Under the same rule the quantity τ is chosen. From Proposition 1 it follows that τ = t/u.

The quantity t is related to the parameters a + , a -, b and c of the tetrahedron T in the following way:

Lemma 5 Let T be a Z 2 -symmetric spherical tetrahedron with dihedral an- gles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Then (i) a 2 --t 2 = a 6 - (s ⋆ 00 ) 2 ∆ ⋆ , (ii) a 2 + -t 2 = a 6 - (s ⋆ 01 ) 2 ∆ ⋆ , (iii) b 2 -t 2 = a 6 - (s ⋆ 02 ) 2 ∆ ⋆ , (iv) c 2 -t 2 = a 6 - (s ⋆ 03 ) 2 ∆ ⋆ ;
where s ⋆ ij , i, j = 0, 1, 2, 3 are respective cofactors of the matrix

G ⋆ s =      1 a + a - b a - c a - a + a - 1 c a - b a - b a - c a - 1 a + a - c a - b a - a + a - 1      .
Proof. Substitute the expression for t 2 from above and proceed with straightforward computations.

The following proposition is used to determine signs of the respective cofactors c ij and c ⋆ ij for i, j = 0, 1, 2, 3 of the matrices G = {g ij } 3 i,j=0 and

G ⋆ = {g ⋆ ij } 3 i,j=0
depending on the signs of their entries:

Proposition 3
The following inequalities hold between entries and cofactors of Gram and edge matrices for a spherical tetrahedron T:

• g ij c ⋆ ij ≥ 0, • g ⋆ ij c ij ≥ 0;
where i, j = 0, 1, 2, 3.

Proof. By [2, Ch. 1, § 4.2] we have

g ij = c ⋆ ij c ⋆ ii c ⋆ jj , g ⋆ ij = c ij √ c ii c jj and c ii > 0, c ⋆
ii > 0 for i, j = 0, 1, 2, 3. Thus

g ij c ⋆ ij = (c ⋆ ij ) 2 c ⋆ ii c ⋆ jj ≥ 0, g ⋆ ij c ij = c 2 ij √ c ii c jj ≥ 0,
where i, j = 0, 1, 2, 3.

The following Lemma provides some useful identities that are used below:

Lemma 6 The following equalities hold:

(i) Re sinh -1 (x) + Re sinh -1 (y) = Re sinh -1 (x y 2 -1 + y √ x 2 -1), where x, y ∈ iR, Im x, Im y ≥ 0, (ii) Re sinh -1 (x) -Re sinh -1 (y) = Re sinh -1 (-x y 2 -1 + y √ x 2 -1), where x, y ∈ iR, Im x, Im y ≥ 0, (iii) Re sinh -1 (x) + Re sinh -1 (y) = Re sinh -1 (x y 2 + 1 + y √ x 2 + 1), where x, y ∈ R, x, y ≥ 0, (iv) Re sinh -1 (x) -Re sinh -1 (y) = Re sinh -1 (x y 2 + 1 -y √ x 2 + 1)
, where x, y ∈ R, x, y ≥ 0, Proof. Using the logarithmic representation for the function sinh -1 (•) and properties of the complex logarithm log(•) one derives the statement of the Lemma for the real parts of corresponding expressions.

We need the relations below to derive the volume formulae for a Z 2symmetric spherical tetrahedron. Then the following cases are possible:

(i) if A + D ≥ π, B ≥ π 2 , C ≥ π 2 and t 2 ≤ 0, then Re sinh -1 a + t + sinh -1 b t + sinh -1 c t + sinh -1 a - t = 0. (ii) if A + D ≥ π, B ≤ π 2 , C ≥ π 2 , then t 2 ≥ 0 and Re -sinh -1 a + t + sinh -1 b t -sinh -1 c t -sinh -1 a - t = 0, (iii) if A + D ≥ π, B ≤ π 2 , C ≤ π 2 and t 2 ≤ 0, then Re -sinh -1 a + t + sinh -1 b t + sinh -1 c t -sinh -1 a - t = 0, (iv) if A + D ≤ π, B ≥ π 2 , C ≥ π 2 , then t 2 ≥ 0 and Re sinh -1 a + t -sinh -1 b t -sinh -1 c t -sinh -1 a - t = 0, (v) if A + D ≤ π, B ≤ π 2 , C ≥ π 2 and t 2 ≤ 0, then Re sinh -1 a + t + sinh -1 b t -sinh -1 c t -sinh -1 a - t = 0, (vi) if A + D ≤ π, B ≤ π 2 , C ≤ π 2 , then t 2 ≥ 0 and Re sinh -1 a + t + sinh -1 b t + sinh -1 c t -sinh -1 a - t = 0,
Proof. Consider case (i). For the edge matrix G ⋆ s of the associated symmetric tetrahedron the following equality holds:

c a - s ⋆ 00 + b a - s ⋆ 01 + a + a - s ⋆ 02 + s ⋆ 03 = 0.
Proposition 3 and Lemma 4 imply that

s ⋆ 00 ≥ 0, s ⋆ 01 A + ≤ 0, s ⋆ 02 B ≤ 0, s ⋆ 03 C ≤ 0,
where all the quantities

A + = cos A + D 2 , B = cos B, C = cos C
are non-positive under assumptions of case (i). Meanwhile A -= cos D-A 2 is non-negative. Therefore,

s ⋆ 00 ≥ 0, s ⋆ 01 ≥ 0, s ⋆ 02 ≥ 0, s ⋆ 03 ≥ 0
and, by Lemma 5,

a 2 --t 2 = a 3 - s ⋆ 00 √ ∆ , a 2 + -t 2 = a 3 - s ⋆ 01 √ ∆ , b 2 -t 2 = a 3 - s ⋆ 02 √ ∆ , c 2 -t 2 = a 3 - s ⋆ 03 √ ∆ .
So, the following equality holds:

a + b 2 -t 2 + b a 2 + -t 2 = -c a 2 --t 2 -a -c 2 -t 2 .
Suppose that t = 0. Then the equivalent form of the equality above is b t

a 2 + t 2 -1 + a + t b 2 t 2 -1 = - c t a 2 - t 2 -1 - a - t c 2 t 2 -1.
Applying function sinh -1 (•) to the both sides of the equality above and making use of relation (i) from Lemma 6, one obtains equality (i) of the present Proposition. If t = 0, then the statement holds in the limiting case t → 0. The proof for cases (iii) and (v) follows by analogy.

Considering cases (ii), (iv) and (vi), note that in consequence of the assumptions imposed on parameters A, B, C, D the quantity τ is purely imaginary and Im τ ≥ 0. Then t is also purely imaginary and Im t ≥ 0. The rest of the proof follows by analogy with cases (i), (iii) and (v), making use of Lemma 6.

Proposition 5 Let T be a spherical Z 2 -symmetric with dihedral angles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Without loss of generality, assume that A ≥ D or, equivalently, l A ≤ l D and, furthermore, l B ≥ l C .
Then in cases

(i) ⋆ A + D ≥ π, B ≥ π 2 , C ≥ π 2 and t 2 ≥ 0, (ii) ⋆ A + D ≥ π, B ≤ π 2 , C ≤ π 2 and t 2 ≥ 0, (iii) ⋆ A + D ≤ π, B ≤ π 2 , C ≥ π 2
and t 2 ≥ 0, Proposition 4 holds for the tetrahedron T ⋆ that is dual to the given one.

Proof. By means of the equality

t 2 = 4 (a + a --bc)(a + b -a -c)(a + c -a -b) ∆ , with a + = cos l A +l D 2 , b = cos l B , c = cos l C , a -= cos l D -l A 2
, the parameter t can be real only if not all of the quantities a + , b, c are negative.

Without loss of generality, assume that the following cases are possible:

(i) ⋆⋆ l A + l D ≤ π, l B ≤ π 2 , l C ≤ π 2 , (ii) ⋆⋆ l A + l D ≤ π, l B ≥ π 2 , l C ≥ π 2 , (iii) ⋆⋆ l A + l D ≥ π, l B ≥ π 2 , l C ≤ π 2 .
Each case above implies that the dihedral angles of the dual tetrahedron T ⋆ fall under conditions (i), (iii) or (v) of Proposition 4. Parameter τ ⋆ of the tetrahedron T ⋆ computed from its dihedral angled satisfies the equality (τ ⋆ ) 2 = -t 2 ≤ 0. It implies that the parameter t ⋆ for the dual tetrahedron T ⋆ computed from its edge lengths also satisfies condition (t ⋆ ) 2 ≤ 0.

Thus, the tetrahedron T ⋆ , that is dual to the given one, falls under one of the cases (i), (iii), (v) of Proposition 4.

Volume of a Z 2 -symmetric spherical tetrahedron

Let T be a Z 2 -symmetric spherical tetrahedron with dihedral angles A, B = E, C = F , D and edge lengths l

A , l B = l E , l C = l F , l D . Denote l + A = l A + l D 2 , l - A = l A -l D 2 , A + = A + D 2 , A -= D -A 2 , a + = cos l + A , a -= cos l - A , b = cos l B , c = cos l C .
Recall that the principal parameter u of the tetrahedron T is the positive root of quadratic equation

u 2 + 4 (a + a --bc)(a + b -a -c)(a + c -a -b) ∆ ⋆ = 1, with ∆ ⋆ = (a + + a -+ b + c)(a + + a --b -c)(a + -a --b + c)(a + -a -+ b -c).
The auxiliary parameter t from Proposition 4 satisfies the equality

t 2 = 1 -u 2 = 4 (a + a --bc)(a + b -a -c)(a + c -a -b) ∆ ⋆ .
Without loss of generality, distinguish the following cases:

(i) A + ≥ π 2 , B ≥ π 2 , C ≥ π 2 and t 2 ≤ 0, (i) ⋆ A + ≥ π 2 , B ≥ π 2 , C ≥ π 2 and t 2 ≥ 0, (ii) A + ≥ π 2 , B ≤ π 2 , C ≥ π 2 , (iii) A + ≥ π 2 , B ≤ π 2 , C ≤ π 2 and t 2 ≤ 0, (iii) ⋆ A + ≥ π 2 , B ≤ π 2 , C ≤ π 2 and t 2 ≥ 0, (iv) A + ≤ π 2 , B ≥ π 2 , C ≥ π 2 , (v) A + ≤ π 2 , B ≤ π 2 , C ≥ π 2 and t 2 ≤ 0, (v) ⋆ A + ≤ π 2 , B ≤ π 2 , C ≥ π 2 and t 2 ≥ 0, (vi) A + ≤ π 2 , B ≤ π 2 , C ≤ π 2 . Define the auxiliary function V(ℓ, u) = 1 2 π/2 ℓ Im log 1 -i u 2 / sin 2 σ -1 1 + i u 2 / sin 2 σ -1 dσ.
for all (ℓ, u) ∈ R 2 . The branch cut of log(•) runs from -∞ to 0. The detailed properties of the the function V(•, •) will be specified in the next section. Set

H = π 2 -A + l + A + π 2 -B l B + π 2 -C l C - π 2 -A -l - A and I = sgn π 2 -A + V(l + A , u) + sgn π 2 -B V(l B , u) +sgn π 2 -C V(l C , u) -V(l - A , u),
where sgn(•) means the sign function.

The following theorem takes place: Subject to the condition of the Theorem, there are possible cases (i)-(vi) pointed above. Consider case (i):

π/2 ≤ A + ≤ π, π/2 ≤ B ≤ π, π/2 ≤ C ≤ π.
By the assumption of the Theorem one has 0 ≤ A -≤ π/2. Thus,

I = -V(l + A , u) -V(l B , u) -V(l C , u) -V(l - A , u)
and

H = π 2 -A + l + A + π 2 -B l B + π 2 -C l C - π 2 -A -l - A .
Note that if u ≥ 0 and 0 ≤ ℓ ≤ π then

V(ℓ, u) = π/2 ℓ Re sin -1 sin σ u dσ + π 2 ℓ - π 2 ,
where the branch cut of sin -1 (•) is (-∞, -1) ∪ (1, ∞).

It follows that the considered function equals

Vol

T = I + H + π 2 ,
where

I = - π/2 l + A Re sin -1 sin σ u dσ - π/2 l B Re sin -1 sin σ u dσ - π/2 l C Re sin -1 sin σ u dσ - π/2 l - A Re sin -1 sin σ u dσ -πl + A -πl B -πl C , and
H = A + l + A + Bl B + Cl C -A -l - A = 1 2 Al A + Bl B + Cl C + 1 2 Dl D .
Once we prove dI = -

1 2 Adl A -Bdl B -Cdl C - 1 2 Ddl D it follows that dVol T = 1 2 l A dA + l B dB + l C dC + 1 2 l D dD
and condition (i) is fulfilled.

Compute the partial derivative

∂I ∂l A = - 1 2 sin -1 sin l + A u + 1 2 sin -1 sin l - A u - π 2 + 1 u ∂u ∂l A F(l + A , l B , l C , l - A , u)

Computation of certain volumes

It follows from Lemma 8 of the next section that in case u = 1 the function V(ℓ, u) has rather elementary form. Thus, the volume of a tetrahedron with the principal parameter u = 1 can be represented by the elementary functions. The equality u = 1 also means the same as t = 0, because of the relation t 2 = u 2 -1 and non-negativity of u.

Consider the associated symmetric tetrahedron T s with its auxiliary parameter t s = 0 because of the relation t = a -t s between t and t s from the proof of Lemma 4.

By Lemma 4 one has

t s = 4(ã -bc)( b -ãc)(c -ãb ) ∆ ⋆ ,
where ∆ ⋆ = det G ⋆ s is the determinant of the edge matrix G ⋆ s of the tetrahedron T s . Also the following equalities hold:

ã = cos l α = a + a - , b = cos l β = b a - , c = cos l γ = c a - .
The equality t s = 0 implies three cases: ã -bc = 0, or b -ãc = 0, or c -ãb = 0. Together, these equalities imply either ã = b = c = ±1 and the tetrahedron is T degenerate, or ã = b = c = 0 and both tetrahedra T and T s are isometric to a equilateral tetrahedron with edge length π 2 . Without loss of generality, suppose only two of the equalities above to be hold: b -ãc = 0 and c -ãb = 0. If the tetrahedron T s is not degenerate, then one obtains b = c = 0. Therefore, tetrahedra T s provide a one-parametric family of tetrahedra with 0 < l α < π, l β = l γ = π 2 . The associated tetrahedron T has edge lengths 0 < l A , l D < π, l B = l C = π 2 . Suppose now that only one equality, namely ã -bc = 0, holds. By Lemma 4 and formulae of spherical geometry from [2, Ch. 1, § 4.2] one obtains cos α = -ã, cos β = b, cos γ = c.

Thus, for T the following inequalities hold:

cos l + A cos A + ≤ 0, cos l B cos B ≥ 0, cos l C cos C ≥ 0.
Apply Theorem 1 to the tetrahedron T with principal parameter u = 1 and obtain that sin l + A = sin A + , sin l B = sin B, sin l C = sin C, sin l - A = sin A -.

From the above one derives the following equalities:

A + = π -l + A , B = l B , C = l C , A -= l - A or, equivalently, A = π -l A , B = l B , C = l C , D = π -l D .
The cases of equalities b-ãc = 0 and c-ã b = 0 are analogous. Moreover, they are the same up to a permutation of the parameters l B and l C of the tetrahedron T.

Thus, the other possible equalities are

A = l D , B = π -l B , C = l C , D = l A or A = l D , B = l B , C = π -l C , D = l A .
Note, that the last three considered cases cover all the occasions mentioned above, like equilateral tetrahedron with edge length π 2 or the family of tetrahedra with edge lengths 0 < l A , l D < π, l B = l C = π 2 . The following statement holds:

Proposition 6 Let T be a spherical Z 2 -symmetric tetrahedron. Suppose cos l + A cos l - A -cos l B cos l C = 0, or cos l B cos l - A -cos l + A cos l C = 0, or cos l C cos l - A -cos l + A cos l B = 0.
Then the volume of T is given by the corresponding formula

Vol T = 1 2 - l 2 A 2 + l 2 B + l 2 C - l 2 D 2 , or Vol T = l A l D -l 2 B + l 2 C 2 , or Vol T = l A l D + l 2 B -l 2 C 2 .
Proof. Let us consider the case cos l + A cos l - A -cos l B cos l C = 0. From the consideration above one obtains that A = π -l A , B = l B , C = l C and D = π -l D . The principal parameter u of T satisfies the equality u = 1. Apply Theorem 8 and Lemma 8 to compute the volume of T using elementary functions. Simplifying the corresponding equation one arrives at the statement of the Proposition.

The proof for other cases follows by analogy. Note, that the claims of Proposition 6 on the tetrahedron T imply its associated symmetric tetrahedron T s has at least one right triangle face. The tetrahedron T itself might not have such one.

Properties of the auxiliary function V(ℓ, u)

The list of basic properties which the function

V(ℓ, u) = 1 2 π/2 ℓ
Im log 1 -i u 2 / sin 2 σ -1 1 + i u 2 / sin 2 σ -1 dσ with (ℓ, u) ∈ R 2 enjoys is given below:

Lemma 7 The function V (ℓ, u) defined above satisfies the following properties:

(i) V (ℓ, u) is continuous and a.e. differentiable in R 2 .

For all (ℓ, u) ∈ R 2

(ii) V (ℓ, u) = V (ℓ, -u),

(iii) V (π -ℓ, u) = -V (ℓ, u), (iv) V (ℓ, u) + V (-ℓ, u) = 2 V (0, u).

The function V (ℓ, u) is linear periodic with respect to ℓ, that is (v) V (ℓ + kπ, u) = V (ℓ, u) -2 k V (0, u) for all k ∈ Z.

Proof. The properties (i)-(iii) follow immediately from the definition of the function V(ℓ, u).

To prove (iv) notice that the equality holds if ℓ = 0. In accordance with the definition of V (ℓ, u), the derivatives of both sides of (iv) with respect to ℓ vanish. Thus, the equality holds.

The derivatives of both sides of (v) with respect to ℓ are equal. Verification of the equality for ℓ = 0 results in the complete proof of (v). By (iii) and (iv) it follows that V (kπ + π, u) = -V (-kπ, u), -V (-kπ, u) = V (kπ, u) -2V (0, u), with k ∈ Z. Hence V (kπ + π, u) = V (kπ, u) -2V (0, u) = • • • = V (0, u) -2(k + 1)V (0, u) and equality (v) holds.

For the special value of u = 1 the function V(ℓ, 1) can be expressed by elementary functions: Lemma 8 The function V(ℓ, u) with u = 1, 0 ≤ ℓ ≤ π can be expressed as

V(ℓ, 1) = 1 2 ℓ - π 2 ℓ - π 2 .
Proof. Notice, that if u ≥ 0 and 0 ≤ ℓ ≤ π then

V (ℓ, u) = π/2 ℓ Re sin -1 sin σ u dσ + π 2 ℓ - π 2 .
Put u = 1 and use the equality

sin -1 x = x, if 0 ≤ x ≤ π/2, π -x, if π/2 ≤ x ≤ π.

Figure 1 :

 1 Figure 1: Spherical tetrahedron

Figure 2 :Lemma 1

 21 Figure 2: Z 2 -symmetric spherical tetrahedron

Proposition 4

 4 Let T be a Z 2 -symmetric spherical tetrahedron with dihedral angles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Without loss of generality, assume that l A ≥ l D or, equivalently, D ≥ A and, furthermore, B ≤ C.

Theorem 8

 8 Let T be a Z 2 -symmetric spherical tetrahedron with dihedral angles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Without loss of generality, assume that A ≤ D or, equivalently, l A ≥ l D and, furthermore, B ≤ C. Then in case tetrahedron T satisfies the condition t 2 ≤ 0 its volume is given by the formula Vol T = I -H. Proof. To prove the theorem we need to show that (i) the function Vol T satisfies the Schläfli formula from Theorem 3, (ii) the function Vol T for the tetrahedron T with edge lengths l A = l B = l C = l D = π/2 equals π 2 /8.

Lemma 9 2 with p k = 1 -

 921 It follows thatV (ℓ, 1) = -1/2(ℓ -π/2) 2 , if 0 ≤ ℓ ≤ π/2, 1/2(ℓ -π/2) 2 , if π/2 ≤ ℓ ≤ π.If u ≥ 1 then we have the following The function V(ℓ, u) with u ≥ 1 has the series representationV(ℓ, u) B(sin ℓ; k + 1, 1/2)/B(k + 1, 1/2),where B(•, •) means betafunction and B(•; •, •) means incomplete beta-function. Marina Paškevič Novosibirsk State University 630090, Pirogova str., bld. 2, Novosibirsk, Russia Pashkevich M@mail.ru

* supported by the Swiss National Science Foundation no. 200020-113199/1 and RFBR no. 09-01-00255, RFBR no. 10-01-00642

where

Proposition 4 implies that F(l + A , l B , l C , l - A , u) = 0. Then, by Proposition 1, the following equality hold:

taking into account that

Thus, condition (i) is satisfied.

Compute the function Vol T with l

2 , A -= 0, setting u = 1 as follows from Proposition 1. Then one has Vol T = π 2 /8 and condition (ii) holds. It implies the Theorem for case (i) to be proven.

The proof for cases (ii), (iii), (iv), (v) and (vi) follows by analogy.

In cases (i) ⋆ , (iii) ⋆ and (v) ⋆ the following theorem holds:

Theorem 9 Let T be a spherical Z 2 -symmetric tetrahedron with dihedral angles A, B = E, C = F , D and edge lengths l A , l B = l E , l C = l F , l D . Without loss of generality, assume that A ≥ D or, equivalently, l A ≤ l D and, furthermore, l B ≥ l C . Then, in case tetrahedron T satisfies the condition t 2 ≥ 0, the statement of Theorem 8 holds for the tetrahedron T ⋆ that is dual to the given one.

Proof. The proof follows by analogy with Theorem 8 using Proposition 5 instead of Proposition 4.

To find the volume of a tetrahedron T that respects the conditions of Theorem 9 one may apply Theorem 8 to the dual tetrahedron T ⋆ and then make use of the Sforza formula from Theorem 4.

Proof. Use the following series representation of the integrand in the expression for V (u, φ) with respect to the variable u at the point u = ∞:

Im log 1 -i u 2 / sin 2 σ -1

The representation above holds for all u ∈ [1, ∞) and σ ∈ R. Integrating the series above with respect to σ from φ to π 2 with φ ∈ [0, π] finishes the proof.

If u ≤ 1 then the function V(ℓ, u) has discontinuous second partial derivatives. Their points of discontinuity in the set {(φ, u) ∈ [0, π] × (0, 1)} are π 2 ± ( π 2 -sin -1 u), u .