Under the common method in which public ways are laid out, altered, widened, or discontinued, the whole amount actually expended in making the improvement is ordinarily borne by the public. There may be voluntary private contributions reducing the amount of the public charge, and such contributions do not invalidate the acts of the authorities charged with the duty of adjudging whether the improvement shall be made, and its details. Parks v. Boston, 8 Pick. 218. Freetown v. Bristol, 9 Pick. 46. Copeland v. Packard, 16 Pick. 217. Crocket v. Boston, 5 Cush. 182. Arlington v. Cutter, 114 Mass. 344. Dudley v. Cilley, 5 N. H. 558. Townsend v. Hoyle, 20 Conn. 1. Dillon, Mun. Corp. (4th ed.) §§ 458, 596. Aside from such contributions, the public ordinarily bears the whole expense actually paid out, and looks only to general taxation for reimbursement. If there are lands which receive from the improvement a benefit or advantage beyond the general advantages to all real estate in the city or town, such peculiar benefit is used to offset and reduce the damage done by taking some part of the land, or injuring it in any other manner, but can be availed of by the public in no other way, and can in no event be availed of to reimburse the public for any expenditure actually paid out in making the improvement.
*243In the development of the country, and particularly in the growth of towns and cities, owners of considerable tracts of land often find it for their pecuniary advantage to lay out and construct streets wholly at their own cost, being more than made good by the increased value of their remaining lands, due wholly to the making of the way.
The theory of our provisions of law authorizing the- assessment of betterments, now contained in Pub. Sts. c. 51, and some additional statutes, (see Sts. 1882, c. 154, § 7; 1884, cc. 226, 237, 280 ; 1885, c. 299 ; 1886, c. 210 ; 1887, c. 124 ; 1891, c. 170, § 3 ; 1893, c. 300, § 2; 1894, c. 288, § 2; 1896, c. 158,) is that in some instances it is just that the public should be reimbursed in whole or in part for the expense of a public improvement by the owners of lands which it peculiarly benefits, limiting the maximum contribution from any particular owner to an amount well within that of his own peculiar or special benefit which he receives in the increased value of his land.
As the instances in which such a forced contribution to the cost of a public improvement is just are frequent in places containing a large population, the betterment laws are in force in cities, while in towns they are not in force unless accepted by the inhabitants. But as it would not be wise or just to have all such improvements made under the betterment system, the authorities charged with deciding whether the improvement shall be made, when they adjudge that it shall be made, determine whether it shall be done under the betterment acts, or in the ordinary way.
If done under the betterment acts, the general plan of enforcing the contribution from the individual landowner is to ascertain, after the completion of the improvement, its whole cost, including full land damages fixed at the value of the land before the making of the improvement, deducting only the value of materials removed by the owner, or of parts of buildings remaining when the value of buildings is allowed for, The peculiar benefit and advantage to every parcel of land which receives from the improvement a benefit or advantage beyond the general advantages to all real estate in the city or town is then adjudged by the authorities who ordered the improvement, and a proportional share of the whole expense of the improve*244ment is assessed by the same authorities upon each parcel of land so peculiarly benefited, with the restriction that no such assessment shall exceed one half the amount of such adjudged benefit and advantage. Thus no person can be forced to contribute in this way more than one half of the peculiar benefit which has been given him by the public in making the improvement.
In the practical working of the system, if the peculiar benefits conferred by the improvement upon individual landowners are more than twice as much as its cost, or just twice as much, its whole cost may be reimbursed to the public by these forced contributions, and nothing borne by general taxation. If the peculiar benefits are less than twice the cost of the improvement, a deficit will remain to be paid out of the proceeds of general taxation.
It is obvious that large powers rest in the hands of those who administer the system. They are to determine whether the cost shall be defrayed by taxation upon the whole municipality, or to a large extent by the owners of a comparatively small body of land so situated as to be peculiarly affected in value by the improvement. In their judgment lies also the determination of what lands are so affected, and the amount of the peculiar benefit, and of the whole value of the land so benefited. When these matters of judgment have been determined, the amount of each particular assessment is a matter of computation.
The power of the Legislature to assess upon the abutting estates the whole cost of the improvement' of a street was affirmed in the case of Dorgan v. Boston, 12 Allen, 223, decided in January, 1866, the case having arisen under St. 1865, c. 159, requiring the aldermen of Boston to widen and lay out as a street Washington Square and adjoining streets. In the same year a betterment act, St. 1866, c. 174, was made operative in Boston, and was the next year extended to Charlestown. St. 1867, c. 224. The constitutionality of this statute was affirmed by the court in 1870 in Jones v. Boston, 104 Mass. 461, upon the ground that the statute did not authorize the aldermen to select at their pleasure some of the estates benefited by the improvement and lay an assessment upon them, but that the *245assessment was to be laid ratably upon all the abutting estates which received any benefit. Before this decision, the betterment act of Boston had been made applicable by St. 1868, c. 75, to any city whose city council should accept it, and in St. 1871, c. 382, the Legislature put the system in force in all cities, and provided that it should be in force in towns upon its acceptance by their several town meetings.
In the practical application of the system, instances would occur in which the adjudication of land damages by the board which ordered the improvement would be unsatisfactory to the landowner. In such cases the damages might be so large as to more than equal the amount of any betterment assessment which could ultimately be made upon the owner, and the adjustment of the damages by the appellate tribunal might be made only after long litigation. By St. 1884, c. 226, the authorities empowered to locate, lay out, or construct streets, ways, or public parks were authorized, whenever they should take or purchase any land therefor, to make an agreement in writing with the owner that the city or town should assume any betterments assessed upon the remainder of his land, or any portion thereof, if he should in turn release upon such terms as might be agreed upon with them all claims for damages.
The authority given in this statute was used in the proceedings out of which grew the present petitions, and it is contended by the petitioners that the results are such as to make the assessment of betterments invalid, and require it to be quashed.
When an agreement is made under St. 1884, c. 226, while assessments are made upon all estates benefited, the assessment upon each owner with whom such an agreement has been made is not collected of him, but is assumed by the city or town. This leaves those owners only with whom no such agreement has been made to pay to the city or town a contribution to the cost of the improvement out of their special benefits, while they must further contribute in their general taxes to so much of the cost as is not met by the betterment assessments not assumed by the city or town. The results of such a system are unfair and unequal, unless in every instance in which such an agreement is made with a landowner its terms are such that in the release of his land damages the town or city gets an equivalent *246for its assumption of the betterment to be assessed upon his remaining land. If it does get such an equivalent, the town or city is not injured, and the landowners with whom no such agreements are made are not injured, for they are assessed no move than their ratable proportion of the cost of the improvement, and the aggregate of that cost is reduced by the release of the land damages of the owners with whom such agreements are made, which land damages would otherwise be included in the cost of the improvement.
While the effect of such an agreement, in which an equivalent for the assumption of the betterment assessment is given in the landowner’s release of damages, injures neither the city or town nor those landowners whose betterment assessments must still be paid, if less than an equivalent is so given, both the city or town and those landowners who are still required to pay their betterment assessments are injured. Such landowners are injured because they are required to make a contribution out of their benefits, when others similarly affected by the improvement are not required to make ratable contributions, but lesser ones, equal only to their land damages, which in the case supposed are not an equivalent for their betterment assessments assumed by the municipality. The town or city is injured, because it gives up its right to collect the assessment which it assumes for less than an equivalent, and, instead of being reimbursed for the cost of the improvement by the owners of the lands specially benefited, must raise the loss by general taxation.
Another circumstance upon which the petitioners rely to defeat the assessment must be noticed in this connection. Agreements under St. 1884, c. 226, were made with many of the landowners, by which they agreed to contribute and pay to the city one third of the cost of construction, “ being credited, however, with the betterments assessed on land of those abutters who do not sign this proposal,” and further agreed to save the city harmless from damages which any owner who does not sign the offer may recover, “ upon being subrogated to and credited with the betterments assessed or to be assessed by said city on such owners respectively.” The expense upon which the assessment was based is an aggregate which has been arrived at not *247taking into account these so called contributions, although a large sum has been actually paid to the city on account of the agreements to contribute toward the cost of construction, and other large sums are due under these clauses of the agreements. The petitioners contend that the expense which is to be made the basis of an assessment of a betterment tax, where contributions by private persons are made to the city in aid of the public improvement, can be only the net expense to the city after deducting the amounts so contributed, and that, because no deduction has been made in the present instance, the basis of the assessment is wrong, and the tax invalid.
Assuming that this contention would be correct in a case where the contribution was a pure gift, we regard the contributions in the present case not as gifts, but as considerations entering into the agreements by which the contributing landowners released their damages, and the city assumed their betterment assessments, and which, in the judgment of the authorities who made the agreements, made the release of the damages and the promised contributions an equivalent to the assumption of the assessments by the city. So regarded, they ought not to be deducted as voluntary gifts in ascertaining the expense which is to be the basis of the betterment assessment, because they are in substance payments made by the landowner’s on account of what would otherwise be their betterment assessments. In place of receiving their land damages and paying their full betterment assessments, they release their damages and pay the difference between the amount of their damages and their asessments by paying a part of the cost of construction and a part of the land damages recovered by other owners, and the city receives no gift to lessen the expense which is the basis of the assessment. We think, therefore, that the question whether the expense which is to be the basis of the assessment was wrongly arrived at in not crediting these contributions because they were gifts does not arise in -the present cases.
Recurring to the effect of the agreements made under St. 1884, c. 226, a majority of the court is of opinion that they should not be held to invalidate the assessment. If in any case there should be a corrupt agreement between a landowner *248and the authorities by which his assessment should be assumed by the city for no adequate consideration, or if by error of judgment a like result should follow, and in either case another landowner should show that he was thus made to contribute to the cost of an improvement when other landowners similarly affected were released from contribution, the assessment might well be quashed. But in the present case there is no allegation of fraud, and no allegation that the agreements under St. 1884, c. 226, were unfair or unjust, either as affecting the city or as causing the petitioners to be assessed sums which in the proper administration of the law they would not be called upon to pay. We are therefore to assume that all parties have acted honestly and fairly, and that the agreements cannot be successfully attacked as one-sided and unreasonably favorable to the landowners whose assessments have been assumed by the city. Taking the figures shown in the documents annexed to the return, computations may be made which apparently show that the city will pay from its general taxes a much greater sum for this improvement than if it had simply paid the land damages and collected all the betterment assessments. This may be explained by supposing that the original awards of land damages were far too low. However this may be, it is not shown that the petitioners are injuriously affected by the agreements made under St. 1884, c. 226.
The petitioners further contend that the assessment is invalid because under the agreements referred to the betterments assessed upon the other landowners are really assessed for the use, not of the city, but of the landowners who entered into the agreements. It is true that taxes cannot be laid by public authorities for the benefit of private persons; and if such were the operation of these agreements, the assessments would be invalid. But such is not their operation. The assessments not assumed by the city are to be paid to the city and to remain its property, like the proceeds of any other taxes.
The expressions in the agreements, that the other landowners are to be credited with the betterments assessed on land of those who do not sign the proposal, and that they are to save the city harmless from land damages upon being subrogated to and credited with the betterments assessed on the owners who may re*249cover damages, are in effect mere limitations of the amounts which the promisors are to pay to equalize the assumption of their betterment assessments. The stipulation of the city is merely that amounts so to be ascertained shall be deducted from one third the cost of construction and the damages recovered in ascertaining what sum is due the city from the promisor under his agreement to release his land damages, in order to equalize that release with the assumption of his assessment by the city.
It is also contended that the assessment is invalid because the original order recited that the expense on which the assessment was based was $112,000, while the assessment amounted to a greater sum, $116,173. But the return shows that the original statement of the expense as $112,000 was a clerical error, after-wards amended by a new order striking out the $112,000 and substituting therefor $117,018, which was the actual expense. This amendment was properly made. Chase v. Springfield, 119 Mass. 556, 563. Grace v. Newton Board of Health, 135 Mass. 490, 498. Foley v. Haverhill, 144 Mass. 352, 355. ISTor was it necessary to give any notice of the amendment. It appears by the cases cited that such an amendment might be made even after the hearing in court upon a petition for a writ of certiorari, and it follows that no earlier notice need be given. The proceeding by amendment was sufficient, and it was not necessaty for the common council to reassess the betterment. If there had been a reassessment for the reason of this clerical error, the amounts assessed would have been the same, and even if the original assessment were void by reason of the error or for want of notice of the amendment, it would be no ground for quashing the assessment upon a writ of certiorari where relief is afforded only when necessary to prevent substantial injustice.
There are other more general considerations which, even if some error or injustice were shown, would justify the denial of relief to these petitioners by quashing the assessment. Under Pub. Sts. c. 51, § 6, parties aggrieved by such an assessment may apply to the Superior Court for a jury to reduce the assessment against them, and they may thus obtain relief if the expense which is the basis of the assessment is wrong, or a disproportionate valuation of their estates has been made. Whiting *250v. Boston, 106 Mass. 89, 97. Prince v. Boston, 111 Mass. 226, 232. Fuller v. Somerville, 136 Mass. 556. Foley v. Haverhill, 144 Mass. 352, 355. These petitioners do not attack the validity of the laying out of the xvay, which has been constructed at great expense and by which they are peculiarly benefited. They have made no contribution to its expense in respect of their peculiar benefits, but have been paid or are entitled to receive their land damages in full, while other parties similarly affected have contributed the damages to their lands. They have not been unaware that the work was being prosecuted under an expectation that betterments should be assessed. One of them was fully aware of all the details of the arrangements which it is now claimed invalidate the assessment,* and the other must be taken to have known of their general nature. Parties who lie by and permit great expenditures to be made, the benefits of which they will enjoy, are not to be allowed to avoid responsibility for the payment of any share of such expenditures by afterwards having the proceedings under which they were incurred quashed. Whately v. County Commissioners, 1 Met. 336. Noyes v. Springfield, 116 Mass. 87. Grace v. Newton Board of Health, 135 Mass. 490, 499.
In the opinion of a majority of the court, the order must be,
Petitions denied.
This is shown by facts recited in the return.